Highly-Efficient Thermoelectronic Conversion of Solar Energy and Heat into Electric Power

1.8 MB·English

by  S. Meir

Highly-Efficient Thermoelectronic Conversion of Solar Energy and Heat into Electric
Power
S. Meir,1 C. Stephanos,1,2 T.H. Geballe,3 and J. Mannhart2,∗
1Center for Electronic Correlations and Magnetism,
Experimental Physics VI, Augsburg University, 86135 Augsburg, Germany
2Max Planck Institute for Solid State Research, 70659 Stuttgart, Germany
3Department of Applied Physics and Laboratory for Advanced Materials,
Stanford University, Stanford, CA 94305-4045, USA
(Dated: 2013-01-15)
Electric power may, in principle, be generated in a highly efficient manner from heat created by
3 focused solar irradiation, chemical combustion, or nuclear decay by means of thermionic energy
1 conversion. Astheconversionefficiencyofthethermionicprocesstendstobedegradedbyelectron
0 space charges, the efficiencies of thermionic generators have amounted to only a fraction of those
2 fundamentally possible. We show that this space-charge problem can be resolved by shaping the
n electric potential distribution of the converter such that the static electron space-charge clouds are
a transformed into an output current. Although the technical development of practical generators
J willrequirefurthersubstantialefforts,weconcludethatahighlyefficienttransformationofheatto
electric power may well be achieved.
5
1

] I. INTRODUCTION thoughtheconversionprocessisstraightforwardandap-
i pears to be achievable: electrons are evaporated from a
c
s Electric power can be generated in a highly efficient heatedemitterelectrodeintovacuum, thentheelectrons
l- manner via thermionic energy conversion from heat cre- drift to the surface of a cooler collector electrode, where
tr ated by focused solar irradiation or combustion of fos- they condense [3, 6]. If used for solar energy harvesting,
m sil fuels [1–4]. Generators based on the thermionic pro- thequantumnatureoflightcanbeexploitedforgreatef-
. cesscould,ifimplemented,considerablyenhancetheeffi- ficiencygainsbyusingphoton-enhancedthermionicemis-
t
a ciency of focused solar energy conversion or of coal com- sion (PETE) [4]. PETE employs the photoeffect to en-
m bustionpowerplants[5], yieldingacorrespondingreduc- hance electron emission by lifting the electron energy in
- tionofCO2 emissions. Inthermionicenergyconversiona a semiconducting emitter across the bandgap ∆ into the
d vacuumisappliedastheactivematerialbetweentheelec- conductionband,fromwheretheelectronsarethermally
n trodes, rather than the solid conductors that give rise to emitted. As a result of the electron flow, the electro-
o
the thermoelectric effect [6]. Thereby, the parasitic heat chemical potentials of the emitter and collector differ by
c
[ conductionfromthehottothecoldelectrodeisradically avoltageVout, andanoutputcurrentIout =Vout/Rl can
decreased. be sourced through a load resistor Rl. Turning this ele-
1 Thermionic generators can operate with input tem- gant operation principle into commercial devices has not
v
5 peratures Tin that are sufficiently high to match the yet been possible, however, because space-charge clouds
0 temperatures at which concentrating-solar power plants suppress the emission current for emitter-collector dis-
5 or fossil-fuel power stations generate heat. In prin- tances of dec > 3–5µm [6, 10, 11]. Practical fabrication
3 ciple, electric power may therefore be generated from of emitter-collector assemblies that operate with the re-
1. these energy sources with outstanding efficiency because quiredclosetolerancesatatemperaturedifferenceTe−Tc
0 the maximum possible efficiency – the Carnot efficiency of many hundred Kelvin was found to be highly chal-
13 gηCene=rat1o−rsoTTuoiunttpu–titnecmrepaesreastuwreit.hInTcino,ntwrhasetr,eaTsoiugtniifiscathnet fleranrgeidngth[1e2rm].aIlnloasdsedsitbioent,wfeoerndeemc i<tte1rµamnd, nceoallre-cfiteolrdbine--
v: amount of energy is wasted today in the conversion of come large [13]. For large dec, it has only been possi-
i heat to electricity. Coal, from which 40% of the worlds ble to suppress the space charges by neutralizing them,
X electricity is currently generated [7], is burned in power which was done by inserting Cs+ ions into the space-
ar stations at ∼1500◦C, whereas, due to technical limita- chargecloud[14,15],amethodusedintwo5kWnuclear-
tions,thesteamturbinesdrivenbythisheatareoperated powered thermionic generators aboard experimental So-
below ∼700◦C, to give but one example. viet satellites [12, 16]. With that approach, compensat-
However, thermionic generators have never been de- ingthespacechargebyioninjectioncausesa∼50%loss
ployed to harvest solar energy or to convert combustion of output power Pout [9]. Novel schemes to suppress the
heat into electricity in power stations [8] or cars [9], al- space charges by optimizing the generation of Cs+ [9]
have yet to be demonstrated. Since the 1950s, when
the space-charge problem was first approached [3, 6, 17],
it has remained the main obstacle to achieving efficient
∗ author to whom correspondence should be addressed, thermionic generators [3, 9].
[email protected]
2
a gate current I by directing them through holes in the
g
gate on helical paths circling straight axes. This pro-
cessturnsthestaticspace-chargecloud,whichpreviously
blocked the electron emission, into a useful output cur-
rent (Fig. 1b,c). The design is analogous to that of ion
thrusters used for spacecraft propulsion.
To investigate the effectiveness of the gate in remov-
ing the space charges, we fabricated a set of thermo-
(a)
electronic generators as model systems (Figs. 2a,b; see
Appendix A). The function of the generators was fur-
thermore modeled by numerical calculations of the elec-
tron emission, space-charge formation and electron tra-
jectories (see Appendix B). Experiment and model cal-
culations provide consistent evidence that, by applying
emitter-gate voltages of Vge∼2–10V, the exact value be-
ing a function of the geometrical design of the genera-
(b)
tor, we can indeed remove the static space-charge clouds
(Figs. 1b,c, 3a). The gate potential enables operation of
thegeneratorsinvacuumwithemitter–collectorspacings
of tens of micrometers (see Fig. 3b).
Although, as will be shown below, the generators op-
erate with high efficiencies at large d , the value of the
ec
emitter–collector current I decreases with d . This
ec ec
is illustrated by Fig. 3b, which shows that the density
Jmax of the emitter–collector current at which the max-
ec
(c) imal output power is obtained, Imax, scales for large d
ec ec
with 1/d2 . At small d , Jmax approaches the current
FIG. 1. Sketch of the working principle of thermoelectronic ec ec ec
density of gate-free generators, because the electric field
generators without (left) and with (right) a gate. The gate,
becomes small inside the mesh holes if d (cid:28) w, where
positivelybiasedwithV =6V,ismountedbetweenemitter ec
ge
w is the grid-mesh diameter defined for hexagonal grids
and collector; a homogeneous magnetic field is applied in x-
direction. (a) Calculated potential profile. (b) Calculated as the distance between opposite corners. For grids with
densityofelectronsinthespace-chargecloud. Theseelectrons finiteconductorwidths, Jmax isfurthermorereducedbe-
ec
donotreachthecollector. (c)Calculateddensityofelectrons cause for t < 1, a fraction of the emitted current is lost
intheemitter–collectorcurrent. Theseelectronsdoreachthe to I . Here, t is the gate transparency, the fraction of
g
collector. The calculations and figures refer to the following the gate area not covered by the conductor. This effect
parameters: φe =2.5eV,φc =0.9eV,Te =1227◦C(1500K), can be minimized by optimizing the gate geometry and
T ≤250◦C, d =100µm, V =(φ −φ )/e, w →0. The
c ec out e c by inducing an inhomogeneous electron emission, for ex-
labels “µ ” and “µ ” refer to the electrochemical potential
e c ample by using nanotubes grown on the emitter. In the
of the emitter and collector; “hν” designates the incoming
latter case, Jmax may be increased further by gate-field-
photons;“c”,“g”,“e”,“v”denotethecollector,gate,emitter, ec
enhanced emission.
and vacuum locations, respectively. The data shown here
were calculated using the 1D model (see Appendix B1). Havingconfirmedthatthespace-chargecloudhasbeen
removed, we now explore the efficiency η = P /P ,
out in
with which these generators transform heat into elec-
II. RESOLVING THE SPACE-CHARGE tric power. The output power of the generator, Pout =
PROBLEM WITH ELECTRIC AND MAGNETIC I V , is maximal for V = (φ − φ )/e, where φ
out out out e c e
FIELDS andφ aretheworkfunctionsoftheemitterandthecol-
c
lector [21], respectively, and e is the elementary charge.
Here we show that the space-charge problem can be For larger Vout, some of the electrons lack the energy to
solved in a plasma-free process. This process involves reach the collector, whereas Iec is independent of Vout
only electrons but no ions. It is therefore best char- for smaller Vout. We start to identify the efficiency limit
acterized as “thermoelectronic”. To remove the static by considering a simplified, ideal case, in which the in-
space charges, a positively charged gate electrode is in- put power Pin is converted completely into an emitter–
sertedintotheemitter–collectorspaceto createa poten- collector current consisting only of electrons at the vac-
tialtrough. Inavirtuallylosslessprocessthistroughac- uum potential (E = 0). If the electrons are only ther-
celeratestheelectronsawayfromtheemittersurfaceand mally emitted, the requirement that the back-emission
decelerates them as they approach the collector (Fig. 1). current from the collector is so small that Imax is pos-
ec
AnominallyhomogeneousmagneticfieldH appliedalong itive entails that η < 1 − Tc (see Appendix B3). To
Te
the electron trajectories prevents loss of the electrons to generate this ideal current, a power of P =Imaxφ /e is
in ec e
3
gate
emitter
collector
1 mm
(a) (b) (c)
FIG.2. (a)Photographofageneratorusedintheseexperiments. Theglowingorangedisk(left)showsthebackoftheresistively
heatedemitter(BaOdispenser);theyellowishdiskedgeontherightshowsthereflectionoftheglowingemitteronthecollector
S. Meir et al., Fig. 2a S. Meir et al., Fig. 2b
surface (steel). (b) Micrograph of a grid (200-µm-thick tungsten foil, w = 0.6mm) used as gate. (c) Setup of a possible
microfabricated generator. The emitter and collector consist of wafers coated with heterostructures (gray lines) designed for
thedesiredworkfunction,thermalandinfraredproperties. Theemitterandcollectorsurfacescomprisenano-hillocksforlocal
fieldenhancements. Thegreenareasmarktheregionsoftheelectronflowthroughthevacuum,thedirectionofI corresponds
out
to the flow of positive charges.
required. Therefore, η =1−φc isastrictupperlimit to 54%, corresponding to a reduction of emissions such
max φe
fortheheat–to–electricpowerconversionefficiency. This as CO2 by ∼17%.
limit also applies to devices in which the photoelectric
effect is used.
In real devices, η is reduced by several loss channels, III. CONCLUSION
whichincludetheabove-neglectedthermalenergycarried
from the emitter by Iemcax, losses due to a finite Ig, radi- Optimizationoftheconversionefficienciesrequiresthe
ation losses from the emitter, thermal conduction of the developmentofmetalorsemiconductorsurfaceswiththe
wirescontactingtheelectrodes,andohmiclosses. Never- desired effective work functions and electron affinities,
theless, only the loss by the electron heat current causes respectively, which may also be done by nanostructuring
a fundamental bound for the efficiency; the other loss theelectrodesurfaces. Thesesurfacesneedtobestableat
effects can in principle be reduced to very small values. hightemperaturesinvacuum. Thetunabilityofthegate
Figure 4 shows the results of the model calculations fieldopenspossibilitiestoaltertheconverterparameters
of the generator efficiencies as a function of the gate during operation. Although the need to generate Cs+
voltage, considering the above-mentioned losses (see Ap- ions to neutralize the space-charge cloud is eliminated,
pendix B3). Starting at V = 0, η increases with V adatoms of elements such as Cs can be used to lower the
ge ge
as the gate potential sweeps the space charges into the work function of the electrodes, in particular of the col-
collector. This increase demonstrates the usefulness of lector. Forhighefficiency, thedevicesmustbethermally
the gate field. At higher V , η decreases because the optimized to minimize heat losses through the wiring.
ge
space charges have been removed and V does not en- Furthermore, thermal radiation of the emitter must be
ge
hance Imax beyond the maximum emission current, but reflected efficiently onto the electrode. For ballistic elec-
ec
increasesthepowerI V lostatthegate. Foragivenφ , tron transport between emitter and collector, a vacuum
g ge e
ηincreaseswithincreasingT duetohigheremissioncur- of better than 0.1mbar is also required, reminiscent of
e
rentsuntilthermalradiationlossesdominate. Forthepa- radio tubes.
rameter range considered realistic for applications (e.g., Such devices may be realized, for example, in a flip-
d = 30µm, t = 0.98, φ = 0.9eV [22]), maximum effi- chip arrangement of oxide-coated wafers separated by
ec c
ciencies of ∼42% are predicted. The calculated efficien- tens of micrometers using thermal-insulation spacers as
cies (Figs. 4) are consistent with previous calculations of sketched in Fig. 2c. This produces hundreds of Watts of
efficiencies of thermionic generators that were presumed power from active areas of some 100cm2. The magnetic
to be devoid of space charges [2, 9, 13, 14, 23]. They fields, typically ≤ 1T with large tolerances in strength
compare well with those of photovoltaic solar cells [20], and spatial distribution, can be generated by permanent
thermoelectric materials [18, 24], and focused solar me- magnets or, for applications such as power plants, by
chanicalgenerators[25,26]. Theresultsoncombinedcy- superconducting coils. Achieving viable, highly efficient
clesshowninFig.4revealthatbyusingthermoelectronic devices requires substantial further materials science ef-
convertersastoppingcyclestheefficiencyofstate-of-the- forts to develop the functional, possibly nanostructured
art coal combustion plants may be increased from 45% materials,aswellasengineeringeffortstoachieveastable
Iout (mA)
4
ACKNOWLEDGMENTS
The authors gratefully acknowledge discussions with
H.Boschker, R.Kneer, T.Kopp, H.Queisser, A.Reller,
H.Ruder, A.Schmehl, and J.Weis as well as technical
supportbyB.FenkandA.Herrnberger. Oneofus(THG)
would like to acknowledge informative conversations on
the use of triodes with longitudinal magnetic fields to
generate Cs plasmas in thermionic generation with the
late Boris Moyzhes, and also acknowledges support for
part of the work at Stanford by the U.S. Department of
Energy, under contract DE-DE-AC02-76SF00515.
(a)
S. Meir et al., Fig. 3a
(b)
FIG. 3. (a) Output current and gate current measured as
a function of V for several gate voltages at T = 1000◦C,
out e
T = 500◦C, w = 1.6mm and d = 700µm. Nominally
c ec
identicalBaOdispensercathodes(φe∼ φc∼ 2.2eV)wereused
for the emitter and collector. (b) Measured and calculated
dependences of Jmax on d . The data was measured at
ec ec
Te = 1100◦C, Tc∼ 500◦C, Vge = 6V; the calculated cur-
rent density refers to the density within the gate mesh. The
output power densities P were calculated from Jmax for
out ec
φ =3eVusingP =Jmax(φ −φ )/e. Theerrorbarsrefer
e out ec e c
to the errors in determining φ , φ , and d . The data for
e c ec
w → 0 and for the curve labeled “without gate” were calcu-
lated using the 1D model including the thermal distribution
of electron velocities (see Appendix B1); the data for w >0
werecalculatedusingthequasi-3Dmodel(seeAppendixB2).
vacuum environment in order to minimize radiative and
conductive heat losses, and to ensure competitive costs.
Remarkably, however, no obstacles of a fundamental na-
ture appear to impede highly efficient power generation
based on thermoelectronic energy converters.
5
FIG.4. Heat–to–electric-powerconversionefficienciescalculatedasafunctionofthegatevoltageofstand-alonethermoelectronic
generators working at a series of emitter temperatures (T =200◦C) and of systems comprising a thermoelectronic generator
c
as topping cycle (d = 30µm). In the combined-cycle systems, the thermoelectronic generators operate between T and
ec e
T = 600◦C. The work functions were selected for optimal performance and T = 1700◦C to allow a comparison with the
s e
efficiency given for the stand-alone system. State-of-the-art steam turbines were presumed to work as bottom cycle, receiving
heat at T and converting it into electricity with η =45%. Owing to the high T of the thermoelectronic generator, φ and
s out e
φ can have rather large values. For the calculation of the efficiencies of the thermoelectronic PETE analogue, a band gap of
c
1.5eVandelectronaffinitiesof1.6and1.85eVwereconsideredforthestand-aloneandthecombined-cyclesystems,respectively
(see Appendix B3). Light–to–electric-power conversion efficiencies for a light-concentration of 5000 are shown for the PETE
systems. TheimagealsoliststheefficienciesofhypotheticalthermoelectricgeneratorswithfiguresofmeritofZT =2and10at
temperatures between T and 200◦C (see [18] and Appendix B3). For comparison, the maximum efficiency of single-junction
in
solar cells is ∼34% (Shockley–Queisser limit [19]) and the best research multi-junction photovoltaic cells have efficiencies of
∼43.5% [20].
6
Appendix A: Experimental Setup and Procedures and
2V d
Inthemodelsystemstheelectrodesweremechanically ϕ (x)=− ge(d −x) for ec ≤x≤d .
g d ec 2 ec
mountedinavacuumchamber(basepressure10−7mbar) ec
to facilitate the study of various converter configura- Atmaximumpoweroutput,emitterandcollectorhave
tions. As emitters, commercial, resistively heated BaO- the same local vacuum potential. We assume the collec-
dispenser cathodes [27] with a temperature-dependent tor to be cold enough that back emission is negligible, as
work function in the range 2.0eV < φe < 2.5eV and an discussed in Ref. [29].
emittingareaof2.8cm2 wereused. Thegateswerelaser- If the thermally distributed initial velocity of emitted
cut tungsten foils, the spacers aluminum oxide foils, and electrons is neglected, the Poisson equation is given by
the collectors either consisted of polished steel plates or
were BaO-dispenser cathodes. The collector work func- J (cid:18) 2e (cid:19)−1/2
tionsweredeterminedfromtheI (V )-characteristics ∆Ψ(x)=− − Ψ(x) ,
out out (cid:15) m
0 e
and additionally from the Richardson-Dushman satura-
tion current. The emitters are ohmically heated, Te whereΨ(x)isthetotalelectrostaticpotentialfornegative
was measured with a pyrometer. The magnetic field charges, consisting of the contribution of the gate and
is generated by two stacks of NdFeB permanent mag- the space-charge potential. This equation is solved ana-
nets mounted on both sides of the emitter-gate-collector lytically, analogous to the Child-Langmuir law [31, 32],
assembly. They created at the gates ∼(200 ± 10)mT. yielding
Photon-enhancement of the emission was not applied.
Electrical measurements were performed with source-
measurement units (Keithley 2400) in 4-wire sensing. (cid:114) e V3/2
ge
J =(cid:15) . (B1)
0 6m d2
e ec
Remarkably, the current density shows the same be-
Appendix B: Model Calculations havior J ∝V3/2/d2 as the Child-Langmuir law.
If the thermal velocity distribution is included, the
Poisson equation becomes
1. One-dimensional models
(cid:20) (cid:21)
en e
∆Ψ(x)=− 0 exp − Ψ(x) ·
For the calculations of the current densities in gate-
(cid:15) k T
0 B
free, plane-parallel configurations the one-dimensional (cid:26) (cid:20) (cid:21)(cid:27)
e
space-charge theory of Langmuir [28] and Hatsopou- · 1±erf (Ψ −Ψ(x)) ,
k T max
los [29] was used to determine the space-charge poten- B
tial. To incorporate the effect of the gate in the one-
where en is the space-charge density at the emitter sur-
dimensionalapproach,thesemodelswereextendedtoin- 0
face and Ψ the maximum of the space-charge poten-
clude the potential generated by an idealized gate, as- max
tial in the inter-electrode space. The plus sign is valid
sumed to be a metal plate that is transparent for elec-
for x ≤ x , the minus sign for x ≥ x , with x
trons and to create a homogeneous electric field. Cal- max max max
being the position of Ψ . n can be determined from
culations of the electric field of a patterned metal grid max 0
the Richardson-Dushman equation [29]; it is a function
withthecommercialelectricfieldsolverCOULOMB[30]
of φ and T . This self-consistent differential equation
showed that for d > w the generated field is virtually e e
ec has to be solved numerically.
identical to the field of an idealized gate. The 3D calcu-
We used Mathematica 8.0 for the numerical calcula-
lations of the electric field distribution and the electron
tions. For each iteration step, the change of the space-
paths in the electric gate field and the applied magnetic
charge potential has to be kept small, as already a small
field done with the commercial software LORENTZ [30]
modification of Ψ(x) can lead to a strong modification
showedthattheelectronsareforcedonquasi-onedimen-
or even a divergence of the solution. Therefore, the so-
sionalpathsbythemagneticfieldandarethuschanneled
lution has to be approached slowly to impede a strong
through the gate openings.
oscillatory behavior.
ToexploreJmaxasafunctionofV belowwecalculate
e ge The model calculations labeled w →0 in Fig. 3b were
the course of the electric potential in the vacuum gap.
obtainedusingtheidealtransparentgatemodelincluding
For this we consider a symmetrical setup, the gate being
the electron velocity distribution.
locatedinthemiddlebetweenemitterandcollector. The
gate potential for electrons is given by
2. The quasi-3-dimensional current tube model
2V d
ϕ (x)=− gex for0≤x≤ ec, To take the inhomogeneities of the electric field of the
g dec 2 gate electrode into account, the interelectrode space was
7
subdivided into narrow prisms, which extend from the by the respective Richardson-Dushman current densities
emitter to the collector surface. We calculated the av- (Eq. B2).
erage gate potential in each prism with the electric field Taking into account the heat transported back to the
solver COULOMB [30]. We apply a linear regression to emitter by the back-emission, the efficiency is obtained
determine the mean electric field, which can be used in to be:
the one-dimensional gate model. We then calculate the
current density for each prism separately. Thereby the
Jmax(φ −φ )
interactions between the prisms were neglected, which η = ec e c . (B4)
Jmax(φ +2k T )−Jmax(φ +2k T )
is a good approximation for the case of small inhomo- e e B e be e B c
geneities in the space-charge density. The total current
This value is known to always be smaller than the
density was obtained by summing up the contributions
Carnot efficiency [3, 17].
from all tubes.
However, the efficiency may be ultimately increased if
Due to the high computational effort required to solve
electronsareemittedonlyatadiscreteenergyE ,sothat
0
the 1D model including the thermally distributed ini-
the 2k T-terms in Eqs. B3 and B4 disappear. For this
B
tial electron velocity, the analytical solution (Eq. B1)
case, however, the Richardson-Dushman equation does
was used to determine the current density, which yields
notapply. Instead,theemittedcurrentdensityhastobe
a good approximation in the voltage range considered.
calculated for a hypothetical material with the discrete
However, it does not account for the temperature-
energy level E , from which the emission of electrons oc-
0
dependence of the current density.
curs. This level may be at or above the vacuum level
E . This calculation can be performed by inserting
vac
a δ-function to describe the discrete density-of-states at
3. Efficiency calculations E = E . In this case, in Eq. B2 no Gaussian-integral
0
has to be determined and the resulting, discrete current
Calculation of the ultimate efficiency limit density J does not have a term with coefficient T2.
E0
As for any thermoelectronic generator, an output
The Richardson-Dushman equation describes the cur- power is only generated for
rent density for electrons emitted from a metal sur-
Jmax =Jmax−Jmax =Jmax−Jmax >0,
face [33]. It is obtained by using the equation J =−nev ec e be e,E0 be,E0
and integrating the Fermi distribution f over all elec-
FD implying
tronswithapositivevelocitynormaltotheemittingsur-
(cid:18) (cid:19) (cid:18) (cid:19)
face, i.e., exp −φe −exp −φc >0.
k T k T
B e B c
(cid:90)(cid:90)(cid:90) em (cid:18) −φ (cid:19) It follows
J =−e d(cid:126)vv f ((cid:126)v)≈ e exp ·
RD vx>0 x FD 4π2(cid:126)3 kBT φc > Tc,
φ T
∞ ∞ ∞ e e
(cid:90) (cid:90) (cid:90) (cid:18)−m v2(cid:19)
· dvx dvy dvzvxexp 2keT = and therefore
B
0 −∞ −∞ Jmax/e·(φ −φ ) φ −φ
(cid:18) (cid:19) η = ec e c = e c =
=−ARDT2exp k−BφT . (B2) =1−Jφemcca<x/1e·−φeTc =η φe.
φ T Carnot
A : Richardson-Dushman constant, φ: work function, e e
RD
T: surface temperature, v: electron velocity. For J →0, it follows
If all non-fundamental channels of heat loss are ne-
φ T
glected, heat is lost from the emitter only by the trans- c → c,
portofelectrons. ThiselectroncoolingPelisgivenby[29] φe Te
and consequently:
∞ ∞ ∞
(cid:90) (cid:90) (cid:90) (cid:18) m v2(cid:19) η →η .
P = dv dv dv v φ+ e f (v)= Carnot
el x y z x 2 FD
As can be seen, the efficiency approaches the Carnot
0 −∞ −∞
J limitifthenetcurrentacrossthevacuumgapapproaches
= ReD(2kBTe+φe). (B3) zero, i.e., if the system approaches equilibrium. Con-
sequently, the output power approaches zero when the
Assuming there is no space-charge cloud limiting the efficiency approaches the Carnot limit. This is a very
transfer of electrons across the vacuum gap, both Jmax typical behavior for any realistic heat engine (see, e.g.,
e
and the back-emission Jmax from the collector are given Refs. [34, 35]).
be
8
Stand-alone generators where the lead is assumed to be metallic and to follow
the Wiedemann-Franz law. With the Lorentz number L
Tocalculatetheefficiencyofrealisticthermoelectronic the thermal conductivity can consequently be expressed
generators, the calculations presented in Refs. [2, 14, 23] as LTmean/Rle. The load is assumed to be at ambient
were extended to include both the gate energy loss and temperature T0. The second term in this equation arises
the dependence of Imax on the gate voltage. In deter- from half of the Joule heat produced in the lead effec-
e
mining the generator efficiency, the power P required tively being transported to the emitter, which can be
g
to sustain the gate electric field is subtracted from the shown by solving the heat flow equation [2].
output power:
Combined-cycle system
P −P
η = out g,
P
in
In combined cycle systems the heat rejected by the
where Pin is the heat input and Pout the power delivered collector (Prej) is used to drive a secondary heat engine
to the load. It is given by workingatanefficiencyofη . Thepowerη P produced
s s rej
by this engine is added to the total produced power,
(cid:18) (cid:19) hence
φ −φ
P = e c −V Imax,
out e lead ec P −P +η P
η = out g s rej.
with the net current flowing to the collector cc Pin
Imax =tImax−Imax, InthesteadystateP isequivalenttotheheattrans-
ec e be rej
portedtothecollector,givenbythesumofanelectronic,
and the voltage drop in the leads connecting the load radiation, and conduction term
with the emitter (R ) and collector (R )
le lc
P =P +P +P ,
V =ImaxR +(Imax−tImax)R . rej elc radc condc
lead ec lc e be le
Here, Imax is the space-charge limited current emitted where
e
fromtheemitter,whichiscalculatedfromthemodelsde-
tImax
scribedaboveandImax theback-emissioncurrentemerg- P = e (φ +Ψ +2k T )−
ing from the collectboer. It has to be considered that Imax elc e c max B e
be Imax
is also reduced by the space-charge potential. Therefore, − be (φ +Ψ +2k T ), (B6)
it is given by e c max B c
P =σ(cid:15)A(tT4−T4),
(cid:18) (cid:19) radc e c
Ψ
Imax =I exp − max ,
be RD kBTc and
iwmituhmthoefRthicehianrtdesr-oenle-DctursohdmeapnotceunrtrieanltΨImRaDx.andthemax- Pcondc = 2RL (Tc−T0)2− (Iemcax2)2Rlc.
In the steady state the heat input equals the sum of lc
all channels of heat loss from the emitter:
P =P +P +P , Losses specific to solar heating
in el rad cond
with the electron cooling:
For solar heated thermoelectronic generators another
Imax fundamental channel for heat loss arises which we take
Pel = ee (φe+Ψmax+2kBTe)− into account: to couple solar radiation into the emitter,
tImax the emitter needs to provide a highly absorbing surface
− bee (φe+Ψmax+2kBTc), (B5) Ab (here “b” stands for black). This surface Ab has a
high emissivity and therefore emits a thermal power P .
b
the radiation cooling: The resulting, reduced light–to–electricity efficiency η is
l
expressed in terms of the heat–to–electricity efficiency η:
P =σ(cid:15)A(T4−tT4),
rad e c
σT4
(A: emitterarea,σ: Stefan-Boltzmannconstant,(cid:15)∼0.1: η =(1− e )η,
effective emissivity of the electrode system [14]) and the l cI0
heat conduction across the emitter lead:
where c is the concentration-factor of the incoming solar
L R
P = (T −T )2− le(Imax−tImax)2, radiationontotheabsorbingspotontheemitter[36]and
cond 2Rle e 0 2 e be I0 the intensity of the incoming solar radiation.
9
PETE-efficiencies Intrinsic electronic heat losses
To calculate the efficiency of a PETE device incorpo-
Below, the relative importance of the channels of heat
rating a gate electrode, we first assume a given emitted
loss will be discussed for the peak of the efficiency of the
current density JPETE and emitter temperature T . The
e e 1600◦C-curve shown in Fig. 4. Although the resulting
latter is chosen such that the hypothetical Richardson-
numbers may slightly vary for other configurations, the
Dushmancurrentdensityacrosstheelectron-affinitybar-
ratiosofthedifferentcontributionsremainessentiallythe
rier (E ) is at least 100 times larger than JPETE. For an
a e same.
ideal PETE-device we then expect an electron yield of 1
electron per above-bandgap-photon [4], as photoexcited At the peak of the efficiency of the 1600◦C-curve
electrons can then be assumed to be thermally emitted shown in Fig. 4 the total input power of Pin =
significantly faster than they recombine. 78.1W/cm2 is mainly consumed by the electron cool-
From JPETE, which defines the emission capability of ing of Pel = 67.3W/cm2. Therefrom, 60.0W/cm2 are
e
the emitter, we then calculate the space-charge limited consumed by the emitted electrons to overcome φe and
currentdensityJmax fromthe1Dmodeldescribedabove 7.3W/cm2 arise from the thermally distributed elec-
ec
(taking into account the thermally distributed starting tron velocity (the 2kBT-terms in Eqs. B5 and B6).
velocity of the electrons). This defines the input power The remaining loss splits up between thermal radiation
actually required to maintain a stable emitter tempera- (Prad =7.0W/cm2)andconductionacrosstheleadwires
ture and, consequently, the required incident light con- (Pcond = 3.8W/cm2). In this configuration the system
centration ceff. For the data shown, this typically yields delivers a power of Pout =36.4W/cm2 to the load cycle,
ceff∼ 500. To satisfy the self-consistency, from ceff and while Pg = 4.0W/cm2 are consumed on the gate. The
Jmax we finally calculate the bandgap ∆ that yields resultingnetoutputpowerof32.4W/cm2 correspondsto
ec
therequiredrateofphotoexcitationsintotheconduction an efficiency of η =42%.
band. Efficiency of thermoelectric generators
We assume the chemical potential to be in the middle
between the worst case (middle of the bandgap) and the
For comparison, efficiencies of hypothetical thermo-
best case (bottom of the bandgap). Consequently, the
electric generators are given in Fig. 4. Those were calcu-
emitter work-function is
lated following, e.g., Ref. [24]:
3
φ =E + ∆.
e a 4 √
T 1+ZT −1
η =(1− out)√ .
From φe and Jemcax the efficiencies of both stand-alone Tin 1+ZT +Tout/Tin
and combined-cycle PETE devices can be calculated as
described above.
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