Third generation hot carrier solar cells: paths towards innovative energy contacts structures

Abstract

The hot carrier solar cell is a very promising clean energy technology, with the potential to achieve high conversion yields with constrained costs. Due to the hot carrier effect, the estimation of the achievable voltage needs some theoretical developments. The classical approach is to consider isentropic energy selective contacts, converting the excess of kinetic energy of the hot carriers into electrical potential energy. Here we show the differences between the ideal case of isentropic contacts and the more realistic one, with an output voltage of the cell depending on the transmission function. We particularly emphasize the importance of the transmission function of the contact on both output current and output voltage, modifying thereby the classical view of the output power dependence on the transmission function.

Full text

Third generation hot carrier solar cells: paths towards
innovative energy contacts structures
Fran¸cois Gibellia,b, Anatole Julianb,c, Zacharie Jehl Li Kaob,c, and Jean-Fran¸cois Guillemolesa,b
aIRDEP, 6 quai Watier, Chatou, France
bNextPV LIA, Meguro-ku, Tokyo, Japan
cOkada Lab, RCAST, 4-6-1 Komaba, Meguro-ku, Tokyo, Japan
ABSTRACT
The hot carrier solar cell is a very promising clean energy technology, with the potential to achieve high conversion
yields with constrained costs. Due to the hot carrier effect, the estimation of the achievable voltage needs some
theoretical developments. The classical approach is to consider isentropic energy selective contacts, converting
the excess of kinetic energy of the hot carriers into electrical potential energy. Here we show the differences
between the ideal case of isentropic contacts and the more realistic one, with an output voltage of the cell
depending on the transmission function. We particularly emphasize the importance of the transmission function
of the contact on both output current and output voltage, modifying thereby the classical view of the output
power dependence on the transmission function.
Keywords: Hot carrier solar cell, energy selective contacts, non-isentropic contact, transmission fuction, contact
Seebeck coefficient
1. INTRODUCTION
The global structure of a hot carrier solar cell is a semiconductor enabling the conversion of light into hot carriers
and two energy selective contacts, one for electrons and another one for holes, 1) enabling to convert the excess
of kinetic energy of the hot carriers into electrical potential energy (useful electrical work), and 2) avoiding cold
carriers from the external electrical circuit to refrigerate the hot carriers in the absorber and thereby annihilate
the hot carrier effect. The global structure of the hot carrier solar cell is shown on figure 1(a).
This theoretical design of the hot carrier solar cell has been experimentally confirmed.1,2Moreover experi-
mental work has been done on the energy selective contacts, involving quantum dot structures3–5or quantum
wells.1
Different theoretical and modeling works have been done to investigate the potential efficiency of hot carrier
solar cells, and most of them consider currents through the contact in a Landauer-B¨uttiker formalism involving
the transmission of the contact. For the output voltage, the classical approach of hot carrier solar cell is to
consider the expression given by Ross and Nozik:6
qV =1−TC
THEext + ∆µH
TC
TH
(1)
However, this expression is the result of considering isentropic energy selective contacts, with vanishingly
small contact width, what implies that there is no ouput current within this assumption.
Here we show how it is possible to consider the ouput voltage of the hot carrier solar cell without considering
isentropic energy selective contacts, and how this output voltage depends on the transmission function of the
contacts.
Further author information: send correspondence to J.-F.G.
jf-guillemoles@chimie-paristech.fr
Physics, Simulation, and Photonic Engineering of Photovoltaic Devices V, edited by Alexandre Freundlich,
Laurent Lombez, Masakazu Sugiyama, Proc. of SPIE Vol. 9743, 97430S · © 2016 SPIE
CCC code: 0277-786X/16/$18 · doi: 10.1117/12.2213562
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Iabs Iem
Te
H, T h
HTC
TC
Ie
+
Ie
−
Ih
+
Ih
−
µe
H
µe
C
µh
H
µh
C
E=0
(a) Global description of the hot carrier solar cell, with one hot
carrier absorbing material, two energy selective contacts and two
electrodes, one for each type of carrier.
Te
H, µe
HTC, µe
C
(b) Energy Selective electron
device (ESED) studied in the
thermoelectric community
Figure 1. Hot carrier solar cell and energy selective contact theoretical descriptions
2. THEORY & MODEL
First we will show the relationship between thermionic engines and energy selective contacts for hot carrier solar
cells; after some considerations on the transmission function the global model will be presented.
2.1 Thermoelectricity and energy selective contacts
In thermoelectricity, devices with one isolated energy level enabling transport of charges between two reservoirs
at different temperatures and with different electrochemical potentials have been widely studied. Indeed, this
kind of devices as presented on figure 1(b) are a kind of brownian engines, and are some experimental approach
of Feynman ratchets when the temperature of the hot reservoir is a periodic function of the time.7–10
The thermoelectric community has studied the influence of the shape of the transmission function11–13 and
the influence of the electron spectrum14–16 on the global efficiency of the thermionic device. Global theoretical
performance optimization have been also performed.17,18
Moreover, by considering the travel of one electron from the hot to the cold reservoir and equalling the heat
removed from the hot reservoir and added to the cold reservoir by the travel of this electron, an isentropic
exchange level can be defined as:19,20
E0=µe
CTe
H−µe
HTC
Te
H−TC
(2)
That is, carriers having exactly this energy in one revervoir can undergo a reversible isentropic transport
to the other reservoir through the energy level at E0. This results of the thermoelectric community has been
recently written in hot carrier terms,21 although the global approach is the same.
For the energy selective contact design, the implications are following:
•if the energy level of the contact is above E0, the energy selective contact will work as an energy generator,
whereas
•if the energy level of the contact is below E0, the energy selective contact will work as a refrigerator.
The first immediate implication for hot carrier energy selective contact design is that it is a necessary condition
to have a contact energy level above the electrochemical potential of the hot carriers, but it is not sufficient since
in order to work as a generator, the energy level of the contact has to be above the isentropic exchange level E0.
Four different cases are shown figure 2.
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µe
H
µe
C
Te
HTC
E0
Eext
(a) Eext < µe
H< µe
C<
E0, the device works in a
refrigerator mode
µe
H
µe
C
Te
HTC
E0
Eext
(b) µe
H< Eext < µe
C<
E0, the device works in a
refrigerator mode
µe
H
µe
C
Te
HTC
E0
Eext
(c) µe
H< µe
C< E0=
Eext, the device works
in an isentropic reversible
mode and there is no par-
ticle nor electrical current
µe
H
µe
C
Te
HTC
E0
Eext
(d) µe
H< µe
C< E0<
Eext, the device works as
a generator
Figure 2. Different operation modes of the energy selective contact, according to the relative position of the energy level
Eext.
2.2 The output voltage of the hot carrier solar cell
In the hot carrier community, the ouput voltage of the hot carrier solar cell has been obtained6by considering
isentropic contacts:
qV =1−TC
THEext + ∆µH
TC
TH
(1)
In order to investigate the difference between electrons and holes, on the one hand we have rewritten this
last expression under the isentropic assumption for electrons and on the other hand for holes:
qV =Ee
extηe+µe
H
TC
Te
H
+Eh
extηh+TC
Th
H
µh
H(3)
But this approach is not totally satisfactory at all, because if the contacts are isentropic, there is no output
current of the hot carrier solar cell, and this expression of the voltage cannot be used formally to compute output
power.
The simpliest way to remove the isentropic assumption while keeping analytical expressions is to consider the
Seebeck coefficient Sof the contact.
With the Seebeck coefficient the global output voltage of the hot carrier solar cell writes:
qV =SeTe
Hηe+µe
H+ShTh
Hηh+µh
H(4)
V=Ve+Vh(5)
where
ηe=1 −TC
Te
H
(6)
ηh=1 −TC
Th
H
(7)
are the Carnot efficiencies of each contact considered as a thermodynamical engine.
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1. show clearly the link with thermoelectricity while considering energy selective contacts for hot carrier solar
cells
2. consider the output voltage without assumption about the entropy exchanges in the contacts
3. show the link between the hot carrier solar cell output voltage with the transmission function.
This last consideration, the dependence of the output voltage with the transmission function is shown by
considering the expression of the Seebeck coefficient of the energy selective contact within the Landauer B¨uttiker
formalism:22
Se=−1
qR(Ee−µe
H)T(Ee)
Te
H
∂f e
C(Ee)
∂E edEe+R(Ee−µe
C)T(Ee)
TC
∂f e
C(Ee)
∂E edEe
RT(Ee)∂f e
C(Ee)
∂E edEe+RT(Ee)∂ f e
C(Ee)
∂E edEe(8)
2.3 About the transmission function
Before computing some results to show the differences induced by non-isentropic contacts in comparison to the
classical isentropic contact approach, a reliable transmission function corresponding to real systems has to be
chosen.
In both hot carrier and thermoelectric communities, the most simple transmission function is the boxcar
function, although it is a rough approximation of the transmission function of real systems. A better approach
is to consider Gaussian shaped transmission functions.
By considering the non-equilibrium Green function formalism to compute transport properties of energy
selective contacts, whatever the kind of mesoscopic device with reduced dimensionnality chosen to achieve this
goal (quantum well, wire, dot), the global shape of the transmission function is a Lorentzian function involving
the couplings Γ between the isolated energy level and the energy levels of both hot and cold reservoirs:23
T(E) = ΓRΓL
(E−Eext)2+(ΓR+ΓL)2
4
(9)
The case of non symmetric couplings between the reservoirs seems to be interesting and can be achieved by
introducing a symmetry parameter a, such as ΓR= (1 −a)Γ and ΓL= (1 + a)Γ, where Γ is the symmetric
coupling. Replacing the couplings with these new expressions in Eq. (9) leads to:
T(E) = (1 −a2)Γ2
(E−Eext)2+ Γ2(10)
This last expressions shows that for a given width Γ2of the transmission function, the influence of the
symmetry parameter aon the transmission function is just a decrease of the value of the transmission, without
any energy shift or additionnal broadening. That is why the influence of the symmetry of the couplings did not
require further computational investigations in the frame of this work.
The transmission function of real systems, computed either with a transfer matrix approach or with non-
equilibrium Green’s functions, is close to the lorentzian shape. An example of a transmission function computed
with a transfer matrix approach for AlGaAs quantum well between GaAs barriers1,2,24 is shown on Figure 3b.
Moreover, growth issues can lead to a broadening of the transmission function from a lorentzian to a gaussian
shape.
Finally the model presented in the next section and the computed results have been obtained with successively
a Gaussian and a Lorentzian shape of the transmission function. In order to facilitate the comparison of the
results, the parameters have been chosen to have an identical width at half maximum called δE, around a peak
located at the contact energy Eext.
The different shapes of the transmission functions are shown on figure 3.
Introducing the Seebeck coefficient enables to
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0.55 0.6 0.65 0.7
0
0.2
0.4
0.6
0.8
1
Energy (eV)
Transmission T
Boxcar
Gaussian
Lorentzian
(a) Comparison between boxcar, Gaussian and
Lorentzian shapes of the transmission function,
for a same width at half maximum δE and con-
tact peak energy Eext
0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
Energy (eV)
Transmission T
(b) Transmission function computed with a
transfer matrix approach for a GaAs/Al-
GaAs/GaAs quantum well
Figure 3. Comparison classical analytical functions describing the transmission and that computed with a transfer matrix
approach with real material parameters
µe
H= 0.35eV
µe
C= 0.4eV
Te
H= 1500KTC= 300K
E0= 0.41eV
Eext
δE
Figure 4. Model and parameters used for the computation
2.4 Global model
The model used to see the influence of non isentropy on the global output voltage is shown on figure 4. We will
consider to reservoirs, one at Te
H= 1500Kwith an electrochemical potential at 0.35eV, and one cold reservoir
at TC= 300Kand an electrochemical potential at 0.40eV. The reservoirs are linked together through an energy
level Eext having an energy with of δE. The two reservoirs are thermostats with fixed parameters whatever the
particle flow between them. With these values of temperatures and electrochemical potentials, the isentropic
exchange energy level can be computed with expression (2), and the corresponding value is E0= 0.41eV .
With this model, the internal energy current density ˙
Ubetween the reservoirs, the Seebeck coefficient of the
contacts Seand the output power will be computed:
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˙
U=N
π~ZET(E)fE−µe
H
kBTe
H−fE−µe
C
kBTCdE (11)
Se=−1
qR(Ee−µe
H)T(Ee)
Te
H
∂f e
C(Ee)
∂E edEe+R(Ee−µe
C)T(Ee)
TC
∂f e
C(Ee)
∂E edEe
RT(Ee)∂f e
C(Ee)
∂E edEe+RT(Ee)∂ f e
C(Ee)
∂E edEe(12)
The Seebeck coefficient will enable to compute the ouput voltage of the contact, that is the contribution of
the thermionic engine to the output voltage of the hot carrier solar cell according to expression (4) for electrons
only:
qV e
non−isen =SeTe
Hηe+µe
H(13)
which will be compared to the classical isentropic expression (3):
qV e
isen =Ee
extηe+µe
H
TC
Te
H
(14)
These two expressions of the voltage enable then to compare the values of the output power of this contact,
relative to the isentropic and non-isentropic approaches.
3. RESULTS & DISCUSSION
The internal energy current density, the output voltage and power have been computed for a gaussian and a
lorentzian transmission shape and are shown respectively on figure 5and figure 6; the results are displayed
versus the position of the isolated extraction level Eext and the energetical width of the contact δE. The Seebeck
coefficient and output voltages are shown in volts, whereas the internal energy current density as well as the
output power are given in arbitrary units, because they depend on the number of channels per unit area, taken
here at 1·1016 m−2. Nevertheless these results in arbitrary units can also be compared because they are computed
with the same parameters and only the transmission function shape is changed.
3.1 Gaussian transmission shape
The internal energy current density of the gaussian transmission shape is shown figure 5(a). As discussed in
paragraphs 2.1 and 2.4, the reversible isentropic energy with the parameter chosen for the model is at 0.41eV:
figure 5(a) shows that below this value in the extraction energy, the outgoing internal energy current density is
negative, indicating that the device is working as a refrigerator. Moreover, whereas for a very narrow energy
width δE the energy Eext separating the positive and negative values of the outgoing internal energy current
density is at 0.41eV, this energy depends on the energy width δE and increases with it. This shows a first
limitation of the isentropic approach, even for determining the generator / refrigerator operating modes of the
energy selective contact. If the chosen design of the device is to maximize the internal energy current density,
then Eext = 0.6eV, δ E = 0.3eV gives the highest value.
The Seebeck coefficient of the contact is shown figure 5(b): the Seebeck coefficient corresponding to the
maximal internal energy current density, at Eext = 0.6eV, δE = 0.3eV , is zero. Reminding expression (4), in this
case the wanted voltage increase due to the hot carrier effect will be null, because the Seebeck of the contact
vanishes for the maximal internal energy current density with the parameters chosen for the model. Moreover,
the maximal Seebeck coefficient is obtained for a higher extraction energy (Eext = 0.8eV ) and a lower energy
width (δE = 0.1eV ), corresponding to a lower internal energy current density.
In order to have a deeper insight into this mutual voltage and current dependence on the transmission
function, the voltage computed from the Seebeck coefficient is shown figure 5(c). Classically the output voltage
is considered to depend only on the extraction energy. Figure 5(c) clearly shows that for a given extraction
energy, the output voltage may be very close to it or not, depending on the energy width of the contact.
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0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.3 0.5
δE (eV)
0.7 0.9
-3·1010
-2·1010
-1·1010
0
1·1010
(a) Internal energy current density
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.3 0.5
δE (eV)
0.7 0.9 -0.00015
-0.0001
-5·10−5
0
5·10−5
0.0001
0.00015
0.0002
0.00025
0.0003
(b) Seebeck coefficient
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.3 0.5
δE (eV)
0.7 0.9 0.2
0.3
0.4
0.5
0.6
0.7
(c) Voltage from the Seebeck
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.3 0.5
δE (eV)
0.7 0.9 -0.2
-0.1
0
0.1
0.2
0.3
(d) Voltage difference
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.3 0.5
δE (eV)
0.7 0.9 -8·1009
-6·1009
-4·1009
-2·1009
0
2·1009
4·1009
6·1009
(e) Power in non isentropic case
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.3 0.5
δE (eV)
0.7 0.9 -4·1009
-3·1009
-2·1009
-1·1009
0
1·1009
2·1009
(f) Power difference between non-isentropic and
classical isentropic approaches
Figure 5. Gaussian transmission fonction
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As for the discussion about the Seebeck coefficient, the maximal output voltage is obtained for higher values of
the extraction energy and lower energy widths than in the case of maximizing the internal energy current density.
In order to compare the magnitude of the isentropic assumption, figure 5(d) shows the difference between the
output voltage computed from the Seebeck coefficient and the output voltage computed in the classical isentropic
case. For the region of interest, where the hot carrier effect could be compatible with the parameter chosen for
the model, the approach proposed here is higher by 100-200mV than the result given by the classical isentropic
assumption. Moreover, the negative region for higher contact widths shows that the output voltage classically
computed overestimates the performances of the device because of the isentropic assumption. The region around
Eext = 0.41eV, δ E = 0eV is zero for the difference between the two voltage expressions, exhibiting thereby the
good agreement between the two approaches because only in this region the isentropic assumption is verified.
The output power has been computed from the Seebeck coefficient and the classical Landauer B¨uttiker
expression of the electrical current, and is depicted figure 5(e). It is of interest to note the shift of the maximal
power region on figure 5(e) in comparison to the maximal internal energy current density region on figure
5(a). This shows that the classical wanted condition of maximal power is more compatible with high output
voltage (shown figure 5(c)). Moreover, in order to compare this approach with the classical approach, the power
computed with the classical expression of the voltage has been substracted from the power computed here (figure
5(e)) and the results are on figure 5(f). This figure 5(f) shows that the classical expression of the output power
underestimates the output power in the case of low energy widths, even at the isentropic exchange point, where
our approach give higher results, whereas for higher energy width the classical expression of the output power
overestimates the performance.
From this analysis on gaussian shaped transmission functions of energy selective contacts, the discrepancy
between classically computed output voltage and power has been exhibited as well as the need to take the contact
width into account for computing both output voltage and electrical current.
The next section will enable to see the influence of the contact transmission shape by comparison.
3.2 Lorentzian transmission shape
First the internal energy current density is shown figure 6(a): the result is almost the same as for the gaussian
shape (figure 5(a)), especially for the region exhibiting the maximal internal energy current density. However the
value of the internal current density in the lorentzian case (figure 6(a)) is half that in the gaussian case (figure
5(a)), mainly due to the broadening of the lorentzian function below the half maximum, as shown figure 3.
The output voltage of figure 6(b) is minimal in the region of maximal energy current density, and the global
shape is very different from that with a gaussian shape (figure 5(c)), especially the output voltage range which
is smaller in the lorentzian case. This difference changes also the shape of the voltage difference of figure 6(c),
having an opposite trend to figure 5(d) in the case of a gaussian shape. For the lorentzian case, the isentropic
expression overestimates by 200mV the output voltage in the maximum internal energy current density, whereas
it was underestimated in the gaussian case.
The output power of figure 6(d) is shifted to lower energy widths δE in comparison to the gaussian case
(figure 5(e)), and the power difference in the lorentzian case, shown figure 6(e), exhibits a very narrow region of
positive values around Eext = 0.52eV, δE = 0.1eV . So in all other regions the classical expression overestimates
the output power in the case of a lorentzian shaped contact transmission function.
4. CONCLUSION
Starting from the isentropic assumption behind the output voltage of the hot carrier solar cell, we have derived
two expressions: one isentropic one accounting for electron-hole assymmetry and another one removing the
isentropic assumption by introducing the Seebeck coefficient of the contact.
Moreover, different results of the thermionic literature have been used to show the thermoelectric side of hot
carrier solar cell energy selective contacts.
This approach has enabled us to model the output voltage and power induced by the energy selective contacts,
and thereby to show the importance of taking into account the non-isentropy of these contacts. We especially want
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0
0.2
0.4
Eext (eV)
0.6
0.8
0.05 0.25
δE (eV)
0.45
-3.5·1010
-3·1010
-2.5·1010
-2·1010
-1.5·1010
-1·1010
-5·1009
0
5·1009
1·1010
(a) Internal energy current density
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.25
δE (eV)
0.4
0.36
0.37
0.38
0.39
0.4
0.41
0.42
0.43
0.44
0.45
(b) Voltage from the Seebeck
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.25
δE (eV)
0.4
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
(c) Voltage difference
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.25
δE (eV)
0.4
-1.4·1010
-1.2·1010
-·1010
-8·1009
-·1009
-4·1009
-2·1009
0
2·1009
4·1009
(d) Power in non isentropic case
0
0.2
0.4
Eext (eV)
0.6
0.8
0.1 0.25
δE (eV)
0.4
-7·1009
-6·1009
-5·1009
-4·1009
-·1009
-2·1009
-1·1009
0
(e) Power difference between non-isentropic and
classical isentropic approaches
Figure 6. Lorentzian transmission function
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to emphasize the dependence of the output voltage of the hot carrier solar cell on the shape of the transmission
function of the contacts.
Last this work sheds a new light on the hot carrier solar cell energy selective contacts and could enable to
use thermoelectric experimentally studied structures to develop new energy selective contacts architectures.
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