**2. Theory**

Superlattices (large number of periods with thin quantum wells) in the intrinsic region of p-in cells provide photo-carrier separation (electron-hole pairs or EHP's) via carrier drifting, leading to effective mass separation of photo-carriers and recombination loss reduction; the advantage of using quantum wells in the intrinsic region of the host material (here GaAs) is the widening of the gap of low-gap material (here Ge); the optical gap (Ee1 – Ehh1) [4] caused by the narrow gap layer in the cell, can in essence be tuned to equal-energy incident solar wavelengths. This means that excess carriers may be trapped in quantum wells and thus thermionic escape will lead to *excess* currents in the cell. A cell design that could lead to excess current may occur in an all-GaAs p-i-n cell with a lattice-matched GaAs-Ge superlattice, grown in the middle of the intrinsic region (Figure 2b), and where excess photo-electrons are *thermionically* reaching the conduction band and are swept away by the built-in electrostatic field. In this chapter we provide a first-principles derivation of excess thermionic current, after radiative recombination losses are taken into account. We propose a short superlattice implanted in the middle of a bulk p-i-n solar cell (Figure 2a), with the expectation of excess current generation via concentrated optical illumination. Such a superlattice will be expected to trap photo-carriers with the chance of overcoming the potential barrier. Once such carriers are out of the quantum traps, they will be likely to join the flux of bulk currents in the p-i-n device. These carriers will essentially be swept away by the built-in electrostatic field, espe‐

GaAs p-i-n cell with a lattice-matched GaAs-Ge superlattice, grown in the middle of the intrinsic region (Figure 2b), and where excess photo-electrons are *thermionically* reaching the conduction band and are swept away by the built-in electrostatic field. In this chapter we provide a firstprinciples derivation of excess thermionic current, after radiative recombination losses are taken into account. We propose a short superlattice implanted in the middle of a bulk p-i-n solar cell (Figure 2a), with the expectation of excess current generation via concentrated optical illumination. Such a superlattice will be expected to trap photo-carriers with the chance of overcoming the potential barrier. Once such carriers are out of the quantum traps, they will be likely to join the flux of bulk currents in the p-i-n device. These carriers will essentially be swept

away by the built-in electrostatic field, especially in the mid-region of the structure.

**(a) (b)**

**GaAs** 

**X suns**

**(Jth)**

**Holes** 

the SL (0.5 μm) portion illuminated at concentrated light.

**intrinsic region with the SL (0.5 m) portion illuminated at concentrated light.** 

**Figure 2: (a) A portion of a superlattice embedded in the intrinsic region and illuminated at X suns. Carriers are excited in the quantum wells and thermionically escape to the conduction band (b) Bird's eye view of the** 

**Figure 2.** a) A portion of a superlattice embedded in the intrinsic region and illuminated at X suns. Carriers are excited in the quantum wells and thermionically escape to the conduction band (b) Bird's eye view of the intrinsic region with

**1eV Optical gap** 

**(J+ Jth)**

~0.25m

~10m

cially in the mid-region of the structure.

182 Solar Cells - New Approaches and Reviews

In this section we derive thermionic currents for a single quantum well. The model as shown in Figure 1 includes a double heterostructure with a quantum well formed by a low-gap medium. According to this figure, electrons may be trapped in eigen-states in the quantum wells under both dark or illumination conditions (as long as incident photons have sufficient energy to excite carriers from the valence band to the conduction band). Once in the quantum well, electrons may thermionically acquire energy to overcome the potential barrier ΔEc, formed by the wide-gap medium. The potential barrier is the conduction band difference Ec2 – Ec1=ΔΕc (Fig-2) of the two layers that are in contact. Standard thermionic emission models [2, 3] describe thermal currents due to electrons having sufficient kinetic energy (KE) to overcome the highest conduction band edge Ec2 (KE=E-Ec2 ≥ ΔEc) [3]. Thermionic currents can be described by integrating over all energies above Ec2 through the following expression:

$$J = q \int dE \mathbf{g}(E) f(E) \mathbf{v}(E) \tag{1}$$

Where q is the electronic charge, g(E) is the electronic density of states (DOS) (eV-1 cm-3) of the two-dimensional system (here a quantum well of width Lw), f(E) is the Fermi-Dirac distribution function, and v(E) is the velocity acquired by the escaping electrons to overcome the barrier potential. Expression (1) is usually re-written as

$$J = q \int\_{Ec2}^{\approx} dE \frac{m^\*}{\pi \hbar^2 L\_W} \langle (e^{-(E-E\_T)/kT}) (\frac{2}{m^\*})^{1/2} (E-E\_{C2})^{1/2} \tag{2}$$

Where the two-dimensional DOS was used for quantum wells of width Lw [3], and where the Fermi-Dirac distribution was replaced with a Maxwell-Boltzmann (E-Ec2 >> 3kT). Expression (2) can be written as:

$$\begin{split} J &= \frac{q\sqrt{2m^\*}}{\pi\hbar^2 L\_W} e^{-(E\_{\mathbb{C}2} - E\_{\mathbb{F}})/kT} \int\_{E\_{\mathbb{C}2}}^{\varphi} dE (E - E\_{\mathbb{C}2})^{1/2} e^{-(E - E\_{\mathbb{C}2})/kT} \\ &= \frac{q\sqrt{2m^\* \left(kT\right)^3}}{\pi\hbar^2 L\_W} e^{-(E\_{\mathbb{C}2} - E\_{\mathbb{F}})/kT} \end{split} \tag{3}$$

The latter expression can be written in a more familiar form, in terms of the barrier height ΔEc, which is a distinct property of the superlattice geometry. The exponential term does not change if we add and subtract the lower conduction band EC1. Since

*EC2-EF=(EC2– EC1)+(EC1–EF)*, we can repeat (3) as follows:

$$J = \frac{q\sqrt{2m^\* \left^3}}{\pi\hbar^2 L\_W} e^{-(E\_{\mathbb{C}2} - E\_{\mathbb{C}1})/kT} \times e^{-(E\_{\mathbb{C}1} - E\_{\mathbb{F}})/kT} \tag{4}$$

The last term in (4) is directly related to the total carrier concentration of electrons ntot in region 1 which is the low-gap medium, hence it can be expressed via the conduction band effective density of states Nc (cm-3) and the carrier density according to the standard relation

$$n\_{tot} = N\_{\mathcal{C}} e^{\left(E\_{\mathcal{C}1} - E\_F\right)/kT},$$

with

$$N\_C = 2(\frac{2\,\pi m^\* \, kT}{h^2})^{3/2}$$

Implementing the latter in the current, as in (4), we obtain a finalized form of thermionic current due to thermally escaping electrons from individual quantum wells:

$$\begin{split} J &= \frac{q\sqrt{2m^\*(kT)^3}}{\pi\hbar^2 L\_{\text{VV}}} e^{-(E\_{\text{C2}} - E\_{\text{C1}})/kT} \times \left(\frac{\mathcal{U}\_{\text{tot}}}{N\_{\text{C}}}\right) \\ &= (\frac{qh}{m\_o})(\frac{\mathcal{U}\_{\text{tot}}}{\mu L\_{\text{VV}}}) e^{-\text{AE}\_{\text{C}}/kT} \end{split} \tag{5}$$

Where we replaced the effective mass m\* with the product of rest electron's mass mo and a numerical factor *µ* [mc=(9ml mt 2 ) 1/3, the DOS effective mass] that differs from material to material [4]. The prefactor (*qh/mo*) in (5) is a constant (q for the electronic charge, h for Planck's constant and mo for the electron's rest mass) equal to *1.1644 x 10-22* (A⋅m<sup>2</sup> ); the current becomes:

$$J = 1.1644 \times 10^{-22} \times (\frac{\eta\_{\text{tot}}}{\mu L\_{\text{W}}}) \times e^{-\Lambda \mathcal{E}\_{\text{C}}/kT} \text{ (A / m}^2\text{)}\tag{6}$$

Expression (6) describes thermionic current produced from a quantum well; this current is a strong function of (a) quantum well width (b) barrier height and (c) total carrier density in the well-layer. Total carrier concentration in the well is the sum of (a) dark carrier concentration in the quantum well nqu and of (b) excess photo-carrier concentration in the miniband under illumination, *δnph: ntot=nqu+δnph.*

Hence,

$$J = \frac{1.1644 \times 10^{-22}}{\mu L\_W} \times (n\_{qu} + \delta n\_{ph}) \times e^{-\Lambda E\_\odot / kT} \text{(\$A / m\$^2\$)}\tag{7}$$

The term nqu represents carriers trapped in quantum wells [6]:

2 1 1

2 \*( ) *C C C F E E kT E E kT*

The last term in (4) is directly related to the total carrier concentration of electrons ntot in region 1 which is the low-gap medium, hence it can be expressed via the conduction band effective

> (EC <sup>1</sup>−EF )/kT ,

> > h<sup>2</sup> )

2 1

*W C*

*E E kT tot*

<sup>h</sup> (5)

1/3, the DOS effective mass] that differs from material to material



); the current becomes:

( )/

*C C*

*L N*

Where we replaced the effective mass m\* with the product of rest electron's mass mo and a

[4]. The prefactor (*qh/mo*) in (5) is a constant (q for the electronic charge, h for Planck's constant

<sup>22</sup> / <sup>2</sup> 1.1644 10 ( ) ( / ) *<sup>C</sup> tot kT W <sup>n</sup> <sup>J</sup> e Am* m*L*

Expression (6) describes thermionic current produced from a quantum well; this current is a strong function of (a) quantum well width (b) barrier height and (c) total carrier density in the well-layer. Total carrier concentration in the well is the sum of (a) dark carrier concentration in the quantum well nqu and of (b) excess photo-carrier concentration in the miniband under

> <sup>22</sup> 1.1644 10 / <sup>2</sup> ( ) (/ ) *<sup>C</sup> kT qu ph*

d

*J n n e Am*

2 \*( ) ( )


Implementing the latter in the current, as in (4), we obtain a finalized form of thermionic current

3/2

( )/ ( )/

<sup>h</sup> (4)

3

ntot= NCe

NC=2( 2πm \* kT

3

*q m kT <sup>n</sup> J e*

= ´

*tot kT*


*e*

/

*C*

due to thermally escaping electrons from individual quantum wells:

2

( )( )

p

*qh n*

=

mt 2 )

numerical factor *µ* [mc=(9ml

illumination, *δnph: ntot=nqu+δnph.*

Hence,

*m L*

and mo for the electron's rest mass) equal to *1.1644 x 10-22* (A⋅m<sup>2</sup>

*W*

*L*

m

*o W*

m

*q m kT J ee*


density of states Nc (cm-3) and the carrier density according to the standard relation

2

p*L*

184 Solar Cells - New Approaches and Reviews

with

*W*

$$m\_{qu} = \frac{m^\*(kT)}{\pi\hbar^2 L\_W} \ln\left(1 + \exp(-\frac{E\_1 - E\_F}{kT})\right) \tag{8}$$

E1 is the lowest eigen-state included (suitably thin layers will provide only one eigen-solution in the wells). If the energy difference (E1 – EF) is much greater than *kT*, the dark quantum well contribution is essentially (numerically) negligible compared to excess photo-concentration, as seen from (8). The *nqu* term would become significant if the Fermi level stays near the conduction band edge of the low-gap semiconductor; however this would require relatively high n-type doping levels of the quantum trap material, and hence scattering and increased absorption losses. The goal here is to embed a superlattice region in the intrinsic part of the pi-n cell, with undoped (or low doping level) quantum well layer to reduce scattering. On the other hand, illumination would excite electrons from the valence to the conduction band at wavelengths near the gap of the layer. The quantum-well semiconductor is tuned to photons of equal energy and tuned photo-excitation populates the energy band with excess electrons *δnph*; these electrons can be found from the form:

$$
\delta \mathfrak{m}\_{\text{ph}} = \mathfrak{m}\_o e^{-\mathbf{x}/L\_n} + \sqrt{\frac{\Phi\_{\underline{\lambda}}}{B}} e^{-\alpha \mathbf{x}/2} \tag{9}
$$

Relation (9) is the solution of the diffusion equation [7], describing minority photo-electrons induced in quantum wells under illumination (through *Φλ).* The first term in (9) is a diffusion term in the x direction, with electron diffusion length Ln (the solution of the homogeneous diffusion equation) and the second term includes directly the absorption coefficient in the exponential term and indirectly through the generation rate term Φλ; *B* is the coefficient of radiative recombination *B* respectively. More specifically, if the solar photon flux (cm-2 s-1) is *Fph(λ),* the photon generation rate *(cm-3 s-1*) is taken to be: *Φλ(cm-3s-1)=α (1-R)Fph(λ)*, where *R* is the reflectivity of the surface and *α* is the absorption coefficient. At high solar concentration

(e.g. X=400 or for X ≥100)), *no*, *nqu* < < *X Φλ <sup>B</sup>* (by at least two orders of magnitude: no, nqu ~1011-1012 cm-3, (*Χ*Φ/*Β*) 1/2=1014 cm-3), expression (7) becomes:

$$J\_{ph}(A \, / \, m^2) \cong \frac{1.1644 \times 10^{-13}}{\mu L\_W \text{(nm)}} \sqrt{X} \sqrt{\frac{\Phi\_\lambda}{B}} (e^{-\alpha x/2} e^{-\Lambda E\_C/kT}) \tag{10}$$

Where both nqu and no terms have been removed as negligible compared to X-sun photogeneration. On the other hand, *B* represents the cross section of radiation recombination losses (radiation coefficient); we use a nominal value of the order of *10-10 cm3 s-1=10-16m3 s-1* [8]. Currents in (10) depend on (a) width *Lw* of quantum wells (b) carrier effective mass *m\*=µ mo* (c) solar concentration *X* (# of suns) (d) carrier photo-generation Φλ and (e) conduction band disconti‐ nuity *ΔE*. The formula above describes thermionic escape current from an illuminated quantum well of width *Lw* (nm), with one mini-band and tuned to a specific solar wavelength λο. Such a choice creates favorable conditions for absorption of photons at energy Eo(eV)=1.24/ λο(μm); one can deduce the value of carrier generation rate Φλ by dividing the irradiance Irr(W/ m2 ) at the specific wavelength λο, by the energy of the corresponding photon of energy Eo. In other words, the solar photon flux Fph (m-2 s-1) is the ratio *Irr* / *E*(*λo*). By tuning a quantum well at 1eV (see Figure-1), we increase the prospects of photon absorption at *λο=1.24µm* (note also that E (*λο=1.24µm*)=1.24/1.24=1eV), and hence the solar photon flux at *λο* is (see also Figure-1):

$$F\_{\rm ph}(\lambda\_o) = \frac{I\_{rr}}{E(\lambda\_o)} = \frac{I\_{rr}\{W \,/\, m^2 \, /\, nm\}}{E(\lambda\_o)(I / \, nm)} = 3.12 \times 10^{18} \, m^{-2} \,\text{s}^1 \tag{11}$$

Where the irradiance Irr *at* a primary wavelength *λο=1,240.00 nm* is taken equal *to 0.5W/m2* [1]. The carrier generation rate (absorption coefficient is assumed to be ~ 104 cm-1=106 m-1) is:

$$\Phi\_{\lambda o} = a(1-R)F\_{ph}(\lambda\_o) \cong (10^4 \times 10^2) \times 3.13 \times 10^{18} m^{-3} s^1 = 3.13 \times 10^{24} m^{-3} s^{11}$$

[Note also the approximation: a 1240nm photon, impinging on a 5nm quantum well, suffers negligible reflection, hence *R negligible*]. The conduction bandgap term ΔΕ, is the energy discontinuity between GaAs and Ge layers. Numerical values of ΔΕ are strongly dependent on fabrication conditions and crystal orientation. We compromise with a well-balanced value 350 meV for ΔE [9]. Based on numerical values of the relevant parameters, the composite cell will generate excess thermionic currents at X suns and with N superlattice periods:

$$J\_{ph} = 1.557 N \sqrt{X} \text{(}\mu A \text{ /}cm^2\text{)}\tag{12}$$

The exponential term *e-ax/2* in (10) refers to absorbed radiation within the span of single quantum well (several nanometers), while the absorption coefficient is taken to near 10,000 cm-1 [10]. This means that this exponential term will be very close to 1 within the span of one quantum well (~ 0.998) and it is not a significant term in computing expression (10). Expression (12) is now a finalized result of thermionic current density due to electrons escaping from a Gequantum well illuminated at a primary wavelength *λο*. Note also that we assumed no losses of thermionic carriers once they are above EC2 (work in progress; see also expression (3) above, in relation with Figure 2a).

To summarize, we have developed a thermionic current formula based on the following assumptions:


$$\text{If } f\_{\text{ph}} = 1.557 \sqrt{X} \text{ (}mA \left/cm^2\right); \text{ Per low-gap layer}$$

quantum well of width *Lw* (nm), with one mini-band and tuned to a specific solar wavelength λο. Such a choice creates favorable conditions for absorption of photons at energy Eo(eV)=1.24/ λο(μm); one can deduce the value of carrier generation rate Φλ by dividing the irradiance Irr(W/

) at the specific wavelength λο, by the energy of the corresponding photon of energy Eo. In other words, the solar photon flux Fph (m-2 s-1) is the ratio *Irr* / *E*(*λo*). By tuning a quantum well at 1eV (see Figure-1), we increase the prospects of photon absorption at *λο=1.24µm* (note also that E (*λο=1.24µm*)=1.24/1.24=1eV), and hence the solar photon flux at *λο* is (see also Figure-1):


m−3 s

<sup>2</sup> 1.557 ( / ) *ph J N X mA cm* <sup>=</sup> (12)

[1].

m-1) is:

cm-1=106

m−3 s1

1=3.13×1024

2 18 2 1 (/ / ) ( ) 3.12 10

Where the irradiance Irr *at* a primary wavelength *λο=1,240.00 nm* is taken equal *to 0.5W/m2*

[Note also the approximation: a 1240nm photon, impinging on a 5nm quantum well, suffers negligible reflection, hence *R negligible*]. The conduction bandgap term ΔΕ, is the energy discontinuity between GaAs and Ge layers. Numerical values of ΔΕ are strongly dependent on fabrication conditions and crystal orientation. We compromise with a well-balanced value 350 meV for ΔE [9]. Based on numerical values of the relevant parameters, the composite cell

The exponential term *e-ax/2* in (10) refers to absorbed radiation within the span of single quantum well (several nanometers), while the absorption coefficient is taken to near 10,000 cm-1 [10]. This means that this exponential term will be very close to 1 within the span of one quantum well (~ 0.998) and it is not a significant term in computing expression (10). Expression (12) is now a finalized result of thermionic current density due to electrons escaping from a Gequantum well illuminated at a primary wavelength *λο*. Note also that we assumed no losses of thermionic carriers once they are above EC2 (work in progress; see also expression (3) above,

To summarize, we have developed a thermionic current formula based on the following

**a.** Electrons are assumed to have kinetic energy *KE ≥ Ec2 – Ec1*

**b.** Electrons occupy *2-D* states in quantum wells

**c.** Total current is calculated from *dj=q g(E) f(E) v(E) dE*

will generate excess thermionic currents at X suns and with N superlattice periods:

)×3.13×10<sup>18</sup>

*I I W m nm <sup>F</sup> m s*

( ) ( )( / )

 l

*E E J nm*

*rr rr*

l

*o o*

The carrier generation rate (absorption coefficient is assumed to be ~ 104

)≅(104×102

*ph o*

186 Solar Cells - New Approaches and Reviews

Φλo=α(1− R)F ph (λo

in relation with Figure 2a).

assumptions:

l

m2

The figure below indicates solar thermal current dependence on solar concentration (from (11), N=1). N = 1).

**Figure-3: Thermionic current density dependence on solar concentration. 31mA/cm2 excess thermal current is generated per quantum well at 400 suns (tuned at 1eV,** *N = 1, J ~ 1.557X1/2***). Figure 3.** Thermionic current density dependence on solar concentration. 31mA/cm2 excess thermal current is generat‐ ed per quantum well at 400 suns (tuned at 1eV, *N=1, J ~ 1.557X1/2*).

From Figure-3, we see that highly illuminated quantum well layers at specific wavelengths produce high currents (for instance, 31mA/cm2 at 400 suns). Excess short circuit current Jph From Figure-3, we see that highly illuminated quantum well layers at specific wavelengths produce high currents (for instance, 31mA/cm2 at 400 suns). Excess short circuit current Jph increases drastically with photo-concentration as seen from Figure-3. Such rapid rise is expected (the higher the irradiance the more carriers available for thermal escape from the wells). However, drastic increase in current does not (a) affect open circuit voltage and (b) total

efficiency. Open-circuit voltage remains near the one-sun value (X=1), while fill factors reduce. As a result, the efficiency of the hybrid cell will be affected accordingly. The next section deals with the collection efficiency of the hybrid cell (bulk plus superlattice cell).
