Supplementary material for: Negative space charge effects in photonenhanced thermionic emission (PETE) solar converters
Gideon Segev1, Dror Weisman1, Yossi Rosenwaks1 and Abraham Kribus2
1
2
1
1.1
School of Electrical Engineering, Faculty of Engineering, Tel-Aviv University, Tel Aviv 69978, Israel
School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel Aviv 69978, Israel
Analysis
PETE cathode model
Under steady state conditions the difference between the cathode total optical generation and recombination is the net
current: S1
L  G  R  
J em  J rev
q
(S1)
G is the optical generation, R’ is the total non-equilibrium recombination per unit volume, Jem and Jrev are the cathode
and anode emission currents respectively, L is the cathode thickness and q is the fundamental charge. This derivation
assumes that reverse emission electrons, and all electrons that are reflected off energy barriers in the inter electrodes
space, contribute to the conduction band electron population. S1 The emission current is linear with the electron
concentration:
J em  nK PETE
(S2)
   EBC 
K PETE  qvx exp  

K BTC 

(S3)
The reverse current follows the standard formulation for vacuum thermionic converters S2:
   EBA 
J rev  ATA2 exp   A

K BTA 

(S4)
EBC and EBA are the cathode and anode added energy barriers formed by the space charge in the inter electrodes space.
A discussion of how these energy barriers are determined is found below. vx  qK BT / 2 me is the average electron
velocity perpendicular to the surface, me is the electron mass,  is the cathode surface electron affinity,  A is the work
function of the anode, A  120 A / cm K is the Richardson-Dushman constant, and TC and TA are the cathode and
anode temperatures, respectively. Since this work is concerned with the device upper efficiency limits, only radiative
recombination is considered. Under non-equilibrium conditions, the rate of photons emitted from a unit area obeys S3 :
2
2
R  R0
np
neq peq
(S5)
Where:S3,S4
2
R0  3 2
hc


Eg
 h  d  h 
2
 h 
exp 
 1
 K BTC 
(S6)
neq , peq are the equilibrium charge carrier concentrations. The non-equilibrium recombination per unit volume R' is
R' 
1 R0
 np  neq peq 
L neq peq
(S7)
In the presence of SRH and surface recombination with an effective recombination velocity S, the non-equilibrium per
volume recombination term is:
R' 
1 R0
n  p  neq peq   1 S n  neq 
L neq peq
L
(S8)
Assuming equal hole and electrons excess carrier concentration  n  n  neq  p  peq and inserting Eqs. (S2), (S5), (S7),
(S8) into Eq. (S1) yields:

K
R n  peq  


R0
J
K
n 2   PETE  0 eq
 S n    E  E g   rev  PETE  neq   0
neq peq
neq peq
q
q
 q



(S9)
  E  Eg  is the solar photons flux with energies above the cathode band gap. Solving Eq. (S9) for the excess
carriers' concentration  n and inserting the solution in equation (S2) yields the device net current at a specific voltage.
The conversion efficiency is defined as:

max V  J em V   J rev  v   
Psun
(S10)
The cathode temperature is determined according to an energy balance coupled to the charge carriers equations
described above. It is assumed that the anode is perfectly reflective, namely, there is no radiative heat transfer between
the cathode and anode. By assuming an infrared (IR) coupling element in the cathode that allows absorption of the subbandgap photons of the incident radiation as heat, the energy balance on the cathode is:1
Psun  J rev B  2KBTA   PIR  P0  Prad  J em B  2KBTC 
(S11)
The terms J em 2 K BTC , J rev 2 K BTA represent the energy that is carried away by the electrons emitted from the cathode
and anode. B  C  EBC is the energy barrier height for electrons in the cathode.
P0 is the cathode equilibrium radiative recombination loss:S3
2
P0  3 2
hc
 h  d  h 
3


Eg
 h 
exp 
 1
 K BTC 
(S12)
Prad is the enhancement of the radiative recombination loss due to non-equilibrium:S3
 np

Prad  P0 
 1
 neq peq 
(S13)
PIR is the radiation emitted from the IR coupling element, which absorbs all sub-bandgap radiation and is completely
transparent to supra-bandgap photons:
2
PIR  3 2
hc
Eg

0
 h  d  h 
3
e
h
K BTC
(S14)
1
The sum of PIR and P0 is the full-spectrum black body emission at temperature TC.
In order to calculate the conversion efficiency, the particle balance in equation (S1) is coupled to the cathode energy
balance of Eq. (S11).
1.2
Space charge theory adjusted to PETE devices
The negative space charge effect in PETE devices can be analyzed using Langmuir space charge theory. Following
previous studies2,5, the potential distribution within the inter electrodes space is found by solving Vlasov's and Poisson's
equation. The development presented here considers the electrons in the inter electrode space as collisionless gas and
assumes that the space charge is formed primarily by electrons emitted from the cathode i.e. J rev  J em S2,S5. The
motive electrode space,  , is defined by:
  qVES
(S15)
𝑉ES is the electrostatic potential. The motive diagram in the inter-electrode space is analogous to the energy band
diagram in semiconductors, where the motive at the semiconductor-vacuum interface is exactly the vacuum level at this
point. Motive diagrams for thermionic converters can be found in Ref. S2. Error! Reference source not found.Figure
S1 shows schematic motive diagrams of a PETE device.6 x is the position between the two electrodes where the
cathode is at x  0 and the anode is at x  d . The cathode and anode are held at temperature Tc and TA respectively.
The anode is at voltage V with respect to the cathode Fermi level. Figure S1(a) illustrates the saturation point, where
the maximum motive is just outside the cathode. For voltages below this point, the emitted electrons are accelerated
towards the anode while crossing the vacuum spacing. Hence, in this regime, negative space charge effects are not
considered and the current voltage characteristics are as discussed in Ref. S1. Figure S1(b) illustrates the situation
where the maximum motive  m is at a distance of xm within the inter-electrode space. This is the space charge limited
regime, where electrons emitted from the cathode need to overcome an added energy barrier before they can reach the
anode, and the height of the barrier depends on the amount of space charge present in the gap. The energy barrier repels
some of the electrons emitted from the cathode, which do not have sufficient energy, and accelerates them back towards
the cathode. Figure S1(c) shows the critical point, where the maximum motive is at a point just outside the anode. For
voltages above the critical point, the energy barrier for electrons emitted from the cathode is EBC  eV  C  A  ,
which is no longer dependent on the amount of negative space charge present in the inter electrodes gap.
Figure S1: PETE motive diagrams (a) saturation point, (b) space charge limited regime, (c) critical point.S6
The collisionless electrons distribution function is determined by the solution of Vlasov's equation:S7
0.5
 
  m   x   

u vex  x    2
 
3
2
m




2
e

 
 m   x  me ve  x 
 me 
 
f  x, ve   2ne  xm  


 exp 
0.5
2 K BTC   
 K BTC
 2 K BTC 
  m   x   

u vex  x    2
 
me

 
 
x  xm
(S16)
x  xm
u is the step function, xm is the location of the maximum motive, ne  xm  is the number density of electrons at xm ,

 m is the maximum motive. me is the electron mass, ve  x   vex  x   vey2  vez2
2
motive at point x . The electron density is calculated by inserting Eq. (S16) into:

1/2
and   x  is the value of the






ne  x    dvez  dvey  dvex f  x, vex 
(S17)
After integration, the electron density at every point in the space between the cathode ( x  0 ) and anode ( x  d )
follows:


1  erf
    x   
ne  x   ne  xm  exp  m
 
 K BTC  
1  erf


1







x
  2
 m
 
 K T
B C

 


1


2
  m   x   
 
 K T
B C

 


x  xm
(S18)
x  xm
The motive's equation is Poisson's equation converted to energy units:
d 2  x 
dx 2

q 2 ne  x 
(S19)
ò0
Substituting Eq. (S18) into Eq. (S19) and using dimensionless quantities:  
x  xm
 m 
and  
for dimensionless
K BTC
xo
motive and distance respectively, yields the dimensionless Poisson's equation:
   0
   0
1  erf  12

d 
2 2  exp     
1
d
1  erf  2

2
(S20)
x02   0 K BTC / 2e 2 ne  xm  is a scaling factor. The initial conditions for Eq. (S20) are γ  0  0 , γ  0   0 . The solution of
Eq. (S20) does not depend on the parameters of a specific problem and the operating condition. Hence, Eq. (S20) needs
to be solved only once. x0 can be written in an alternative convenient way, as a function of J em :
  2K 3 
x0   0 B 2 
 2me q 
0.25
TC0.75
0.5
J em
(S21)
The relation between the dimensionless distances  C and  A obeys:
 A  C 
2
d
x0
(S22)
Numerical procedure
The numerical procedure is similar to the procedure done in standard thermionic devices S2,S5 and in the previous PETE
model suggested in Ref. S6. However, in order to allow the electron concentration to vary with the operating point, the
PETE particle balance equations from section 1.1 must be coupled to the negative space charge model presented in
section 1.2. When an energy balance is applied, equation (S11) is coupled to the equation set as well. First, we find the
voltages and currents corresponding to the saturation point and the critical point. Then, using these voltages as upper
and lower bounds, we calculate the currents and matching voltages within the space charge limited range. For voltages
below the saturation point or above the critical point, the negative space charge in the gap does not affect the emission
current and the calculation is as in Ref. S1.
2.1
Saturation point
At voltages below the saturation point, the maximum motive is just outside the cathode and C  A  V . Hence, there
is no added energy barrier for the emitted electrons from the cathode, and EBC  0 . The added energy barrier for the
electrons emitted from the anode is determined by the voltage EBA  C  V  A . As a results, the number of electrons
that are emitted from the anode, reach the cathode and contribute to the cathode's conduction band population is also
influenced by the voltage. In most cases this contribution is small. At the saturation point the dimensionless position of
maximum motive with respect to the cathode is C  sat  0 , and the dimensionless position of maximum motive with
respect to the anode  Asat is:
d  2 m q 2 
 A sat V     2 e3 
x0  ò0 kB 
1
4
1/2
J em
 sat V 
3
Tc
d
(S23)
4
The saturation point voltage Vsat is found by using    , the numerical solution of Eq. (S20), and solving numerically:
Vsat 
C   A  γ  A sat Vsat   K BTC
(S24)
q
where ξ Asat  Vsat  is as in Eq. (S23).
2.2
Critical point
At the critical point the maximum motive is just outside the anode. Under these conditions C  A  Vcrit . Hence, the
energy barrier for electrons emitted from the cathode is EBC   A  Vcrit  C . Since there is no extra energy barrier for
crit
the emitted electrons from the anode, EBA  0 . In this case dimensionless position of the maximum motive with
crit
respect to the anode is,  Acrit  0 , and the dimensionless position of the maximum motive with respect to the cathode
is:
 2 me q 2 
d
C crit V       2 3 
x0
 ò0 k B 
1
4
1/2
J em
crit V 
The critical point voltage, Vcrit , is determined by inserting Eq. (S25) into
3
Tc
4
d
(S25)
Vcrit 
C   A  γ C crit Vcrit   K BTC
q
(S26)
and solving numerically for Vcrit .
2.3
Space charge limited regime
The following procedure can be used for each cathode emission current J em between J crit and J sat in order to find the
matching voltage. This procedure is similar to the one given in S2,S5,S6, however it was modified in order to account for
the dependence of the cathode electrons concentration on the operating voltage:
1.
2.
3.
4.
5.
6.
7.
Choose a voltage between Vsat and Vcrit
Choose an initial value for γC
Calculate the emission current, Jem, according to the particle balance in equations (S1-S8)
Calculate the dimensionless distance, ξC(γC), according to the solution of Eq. (S20)
Calculate x0 according to Eq. (S21) and the solution of step 3
Calculate the dimensionless distance, ξA(γC), according to Eq. (S22) and the solution of steps (4-5)
Calculate the maximum dimensionless motive with respect to the anode, γA(ξA), according to the solution of Eq. (18)
and the solution of step 7
8. Calculate the maximum dimensionless motive with respect to the cathode, γC as defined by the motive diagram and
the result of step 7:
C  
C   A   A K BTC  qV
K BTC
(S27)
9. Repeat steps 3-8 with γC calculated in step 8 until reaching convergence.
For current above J sat and below J crit , the space charge does not create an additional barrier, and the corresponding
voltage is calculated using the procedure as was shown in Ref S1. It should be noted that when the negative space
charge effects are negligible, the maximum power point voltage is higher than the voltage at the critical point.
3
Surface and Shockley Reed Hall Effects
Figure S2 shows the current voltage curves of PETE converters subject to negative space charge with several effective
surface and SRH recombination velocities. The simulation parameters are as in Figure 1, the gap width is 2μm. Since
the cathode operates in the PETE regime, increasing the recombination reduces the cathode electrons concentration and
with it the emission current. This effect is most dominant in the space charge limited regime (voltages between 0.37V
and 1.25V for S=0cm/s) because the added recombination prevents the cathode electrons concentration from increasing
with voltage. For effective recombination velocities above 100cm/s the added recombination reduces the emission
current at voltages below the saturation voltage as well. Furthermore, since the emission current changes with the
effective recombination velocity, the voltage range of the space charge limited regime also changes with the effective
recombination velocity. The inset of Figure S2 shows the saturation voltage, Vsat, and critical voltage, Vcrit, as a function
of the effective recombination velocity, S. The emission current reduction with the recombination velocity leads to a
clear reduction in the voltage range of the negative space charge regime. For these reasons the model with no electron
recycling presented in Ref. S6, in which only electrons that are reflected by the electric field in the gap are annihilated,
cannot represent surface recombination effects properly. Since the analysis in this work is based on a net particle
balance, it cannot discriminate between bulk SRH and surface recombination. In order to fully address the combined
effect of surface recombination and negative space charge a one dimensional model is required such as in S8–S10.
Figure S2, Current-voltage characteristics of PETE converters subject to negative space charge with several effective surface and SRH recombination
velocities. The simulation parameters are as in Figure 1, the gap width is 2μm.
4
Nomenclature
Parameter
Description
<vx>
Average electron velocity [cm/sec]
ARD
Richardson constant [A/cm2K2]
c
Speed of light [cm/sec]
d
Distance between cathode and anode [cm]
EBA
Anode added energy barrier [eV]
EBC
Cathode added energy barrier [eV]
Eg
Cathode band gap [eV]
erf
Error function
f(x,ve )
Distribution function of electrons in inter electrodes space [sec3/cm6]
G
Optical generation [1/cm3sec]
h
Planks constant [eV∙sec]
hν
Photon energy [eV]
Jem
Cathode emission current density [A/cm2]
Jem-crit
Cathode emission current density at the critical point [A/cm2]
Jem-sat
Cathode emission current density at the saturation point [A/cm2]
Jrev
Anode emission current density [A/cm2]
Jrev-crit
Anode emission current density at the critical point [A/cm2]
Jrev-sat
Anode emission current density at the saturation point [A/cm2]
KB
Boltzmann's constant [eV/K]
KPETE
Emission current coefficient [A∙cm]
L
Cathode thickness [cm]
me
Electron effective mass [Kg]
n
Cathode electron concentration [cm-3]
ne(x)
Electron density in inter electrodes space [1/cm3]
ne(xm)
Electron density at the maximum motive [1/cm3]
neq
Cathode equilibrium electron concentration [cm-3]
p
Cathode equilibrium hole concentration [cm-3]
P0
Cathode equilibrium radiation [W/cm2]
peq
Cathode equilibrium hole concentration [cm-3]
PIR
Infra-red coupling radiative losses [W/cm2]
Prad
Radiative recombination losses [W/cm2]
Psun
Input solar power flux [W/cm2]
q
Electron charge [C]
R
Non equilibrium radiative recombination [1/cm2sec]
R'
Bulk non equilibrium average recombination [1/cm3sec]
R0
Equilibrium radiative recombination [1/cm2sec]
S
Effective surface and SRH recombination velocity [cm/sec]
TA
Anode temperature[K]
Tc
Cathode temperature[K]
u
Step function
V
Operating voltage [V]
Vcrit
Operating voltage at critical point [V]
ve
Electron velocity in inter electrodes space [cm/sec]
VES
Electrostatic potential [V]
vox
Minimum velocity in the x direction [cm/sec]
Vsat
Operating voltage at saturation point [V]
x
Lateral position between electrodes [cm]
x0
Scaling factor [cm]
xm
Location of maximum motive [cm]
γ
Dimensionless motive
γA
Dimensionless motive at anode surface
γA-cirt
Dimensionless motive at anode surface at critical point
γA-SAT
Dimensionless motive at anode surface at saturation point
γC
Dimensionless motive at cathode surface
γC-cirt
Dimensionless motive at cathode surface at critical point
γC-SAT
Dimensionless motive at cathode surface at saturation point
δn
Access carriers concentration [1/cm3]
ε0
vacuum permittivity [F/cm]
η
PETE conversion efficiency
ξ
Dimensionless distance
ξA
Dimensionless distance at anode surface
ξA-cirt
Dimensionless distance at anode surface at critical point
ξA-SAT
Dimensionless distance at anode surface at saturation point
ξC
Dimensionless distance at cathode surface
ξC-cirt
Dimensionless distance at cathode surface at critical point
ξC-SAT
Dimensionless distance at cathode surface at saturation point
Φ(E>Eg)
Above band gap photon flux [1/cm2sec]
ϕA
Anode work function [eV]
ϕB
Cathode electron emission barrier[eV]
ϕC
Cathode work function [eV]
χ
Cathode electron affinity [eV]
ψ
The motive in the inter electrode space [eV]
ψm
Maximum motive [eV]
References
S1
G. Segev, Y. Rosenwaks, and A. Kribus, Sol. Energy Mater. Sol. Cells 107, 125 (2012).
S2
G.N. Hatsopoulos and E.P. Gyftopoulos, Thermionic Energy Conversion Vol. 1: Processes and Devices (MIT Press,
Cambridge, MA, USA, 1973).
S3
P. Wurfel, Physics of Solar Cells from Principles to New Concepts (Wiley, 2005).
S4
J.W. Schwede, I. Bargatin, D.C. Riley, B.E. Hardin, S.J. Rosenthal, Y. Sun, F. Schmitt, P. Pianetta, R.T. Howe, Z.-X.
Shen, and N. a Melosh, Nat. Mater. 9, 762 (2010).
S5
J.R. Smith, G.L. Bilbro, and R.J. Nemanich, J. Vac. Sci. Technol. B Microelectron. Nanom. Struct. 27, 1132 (2009).
S6
S. Su, Y. Wang, T. Liu, G. Su, and J. Chen, Sol. Energy Mater. Sol. Cells 121, 137 (2014).
S7
I. Langmuir, Phys. Rev. 21, (1923).
S8
G. Segev, Y. Rosenwaks, and A. Kribus, J. Appl. Phys. 044505, (2013).
S9
K. Sahasrabuddhe, J.W. Schwede, I. Bargatin, J. Jean, R.T. Howe, Z.-X. Shen, and N. a. Melosh, J. Appl. Phys. 112,
094907 (2012).
S10
A. Varpula and M. Prunnila, J. Appl. Phys. 112, 44506 (2012).