**3. Device design**

Incorporating a superlattice-layer GaAs/Ge, with 5nm-thick quantum wells, embedded in the middle of the intrinsic region of the bulk cell, will increase the current dramatically but not the open circuit voltage. This current will add to the bulk GaAs-related current of the PV cell, and will affect the open-circuit voltage as well. The bulk cell illuminated at X suns will produce an open-circuit voltage Vocph given by the standard formula:

$$\boldsymbol{V}\_{oc}\boldsymbol{V}\_{oc} = \boldsymbol{V}\_{t}\ln(\frac{\mathbf{X}\mathbf{J}\_{L}}{\mathbf{J}\_{o}}+\mathbf{1}) \equiv \boldsymbol{V}\_{t}\ln(\frac{\mathbf{X}\mathbf{J}\_{L}}{\mathbf{J}\_{o}}) = \boldsymbol{V}\_{oc} + \boldsymbol{V}\_{t}\ln(\mathbf{X})\tag{13}$$

Where the photo-current has been multiplied by the X (suns) factor [1]. Based on the above, net *OC*-voltage will increase by 15mV: Vt *ln* (X)=0.025 ln (400)=0.150V (with Vt=kT/q) and hence Voc(ph)=1.051+0.15=1.201V. We simulate a GaAs p-i-n cell with the following geometry [11]:


$$\text{I.k.}\qquad\text{The fill factor FF is }FF = 1 - \frac{V\_t}{V\_{oc}^{total}}\ln(1 + \frac{V\_m}{V\_t}) - \frac{V\_t}{V\_{oc}^{total}}\tag{12}$$

**l.** Host (bulk) GaAs-cell (no superlattice, X=400) shows 26.26%, through FF and Vm compu‐ tation

$$V\_m = V\_{oc}^{total} - V\_t \ln(1 + \frac{V\_m}{V\_t}) = (V\_{oc} + V\_t \ln X) - V\_t \ln(1 + \frac{V\_m}{V\_t}) \tag{14}$$

$$\eta = (0.85) \frac{(400 \times 25.73) \times (1.201)}{400} = 26.26\,\%$$

Note that open-circuit voltage is Voc(X=400)=Voc(X=1)+Vt ln (400)=1.201V

The efficiency of the hybrid (bulk+superlattice) device is:

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

Incorporating a superlattice-layer GaAs/Ge, with 5nm-thick quantum wells, embedded in the middle of the intrinsic region of the bulk cell, will increase the current dramatically but not the open circuit voltage. This current will add to the bulk GaAs-related current of the PV cell, and will affect the open-circuit voltage as well. The bulk cell illuminated at X suns will produce an

> ln( 1) ln( ) ln( ) *L L oc t t oc t o o XJ XJ VV V VVX J J*

Where the photo-current has been multiplied by the X (suns) factor [1]. Based on the above, net *OC*-voltage will increase by 15mV: Vt *ln* (X)=0.025 ln (400)=0.150V (with Vt=kT/q) and hence Voc(ph)=1.051+0.15=1.201V. We simulate a GaAs p-i-n cell with the following geometry [11]:

**d.** Longer intrinsic region provides excess photo-carriers swept away from the junctions;

**g.** GaAs/Ge multilayer in the intrinsic region (Ge layers at 5nm): total length of SL region for

**i.** PC1D simulations of a host-GaAs p-i-n host cell, (steps (a) through (d)) lead to Jsc=JL=25.73

**j.** Illumination of the composite cell at X-suns will affect the collection in two ways (i) shortcircuit photo-currents and input power will increase by a factor X (ii) OC-voltage will

> *Vm Vt*

)<sup>−</sup> *Vt Voc*

*total* [12]

); efficiency η=(0.94) (25.73) (1.051) /

**e.** Total current is essentially the sum of minority and off-intrinsic region currents

**h.** Illumination of the whole device (including intrinsic region) at X suns (e.g. X=400)

excess minority carriers are developed in both p-and n-regions

, Voc=1.051V, FF=0.94, at X=1, Pin=100mW/cm2

*Voc*

*total* ln(1 +

= +@ = + (13)

with the collection efficiency of the hybrid cell (bulk plus superlattice cell).

open-circuit voltage Vocph given by the standard formula:

'

**3. Device design**

188 Solar Cells - New Approaches and Reviews

**a.** Total device 14μm)

**b.** Length of p-region 2 μm

**f.** Length of n-region 2 μm

mA/cm2

**k.**

Pin=25.42%

**c.** Length of intrinsic region 10 μm

100 periods, 100 x 5=500 nm=0.5 μm

increase as indicated by (12).

The fill factor FF is *FF* =1<sup>−</sup> *Vt*

$$\begin{aligned} \eta &= (FF) \frac{(Xl\_L + J\_{ph})}{XP\_{in}} V\_{oc}^{\cdot} = (FF) \frac{I\_L + (\frac{J\_{ph}}{X})}{P\_{in}} V\_{oc}^{\cdot} \\ \text{Or} \\ \eta &= (FF) \frac{I\_L + (1.557 \, NX^{-1/2})}{P\_{in}} V\_{oc}^{\cdot} \end{aligned} \tag{15}$$

We re-write (15) by splitting it in two parts containing bulk and superlattice regions:

$$\begin{split} \eta &= (FF)\frac{I\_L + (1.557\,\text{N}\text{X}^{-1/2})}{P\_{in}} V\_{oc}^{\cdot} \\ &= (FF)\frac{I\_L V\_{oc}^{\cdot}}{P\_{in}} + (FF)\frac{1.557\,\text{X}^{-1/2}V\_{ov}^{\cdot}}{P\_{in}} \equiv \eta\_{bulk}^{\times} + \delta\eta \end{split} \tag{16}$$

The first term of (16) is the bulk-cell efficiency at X suns, and the second one is the excess collection efficiency due to the superlattice region. Rewriting (16) we get:

$$
\delta\eta = \eta\_{bulk} + \delta\eta = \left(FF\right)\frac{I\_L V\_{ov}}{P\_{in}}\bigg/ + \left(FF\frac{1.55TV\_{oc}}{P\_{in}}N\right)\bigg/ \mathbf{1} + \frac{V\_t}{V\_{oc}}\ln X\}X^{-1/2} \tag{17}
$$

From (16), we predict total efficiency *η* =26.26*%* + 0.079*N* (*%*)=28.63(*%*) (with FF=0.85, X=400, Voc=1.051V, Vt =0.025eV, N=30). On the other hand, collection efficiency at 400 suns for 50 periods will increase efficiency to 30.21%. The advantage of higher efficiency, at greater N values, is depicted in Figure 4 below:

efficiency:

 

 *bulk*

**Figure-4: Collection efficiency vs number N of 5nm-Ge layers in the intrinsic region of a GaAs p-i-n cell. Note that 42% is feasible with 200 periods at 400 suns (XPin = 400mW/cm^2). Figure 4.** Collection efficiency vs number N of 5nm-Ge layers in the intrinsic region of a GaAs p-i-n cell. Note that 42% is feasible with 200 periods at 400 suns (XPin=400mW/cm^2).

Note from Figure 4, that 42% collection efficiency can be expected at 400 suns and at large number of superlattice periods (N ~ 200). By using (11) in (17) we propose the total efficiency as the sum of two conclude by re-writing as the sum of two factors bulk efficiency and excess Note from Figure 4, that 42% collection efficiency can be expected at 400 suns and at large number of superlattice periods (N ~ 200). By using (11) in (17) we propose the total efficiency as the sum of two conclude by re-writing as the sum of two factors bulk efficiency and excess efficiency:

$$
\eta \langle \% \rangle = \eta\_{bulk} + \delta \eta = \eta\_{bulk} + \{1.557 FF\} \times \frac{N X^{-1/2} V\_{ac}}{P\_{in}} \tag{18}
$$

Expression (17) essentially predicts excess efficiency of a cell, enriched with an embedded N-period superlattice, under concentrated power equivalent to X suns. We would also like to see Expression (17) essentially predicts excess efficiency *δη* of a cell, enriched with an embedded N-period superlattice, under concentrated power equivalent to X suns. We would also like to see excess efficiency against X at a fixed N: (17) can be re-written:

excess efficiency against X at a fixed N: (17) can be re-written:

*in*

$$
\eta = \eta\_{\text{bulk}} + \delta \eta = 26.26 + \frac{1.323 \text{(N)}}{P\_{in} \sqrt{\text{X}}} (V\_{oc} + V\_t \ln X) \tag{19}
$$

Note however from (18) that the excess efficiency δη strongly depends on X-1/2 and N. High X "slows down" excess efficiency as shown from Figure-5:

Note from Figure-5 that high X does not necessarily mean better performance; at 100 and 400 suns, 30 SL layers in the intrinsic region will provide 30.88% and 28.63% collection respectively;

Note however from (18) that the excess efficiency strongly depends on X-1/2 and N. High X "slows down" excess efficiency as shown from Figure-5: Solar Cell Efficiency Increase at High Solar Concentration, by Thermionic Escape via Tuned... http://dx.doi.org/10.5772/59039 191

**Figure-5: Efficiency vs number of suns X, with 30 superlattice layers in the intrinsic region. Excess efficiency decreases with X (see (18)). Note higher efficiencies at lower irradiances**. **Figure 5.** Efficiency vs number of suns X, with 30 superlattice layers in the intrinsic region. Excess efficiency decreases with X (see (18)). Note higher efficiencies at lower irradiances.

**Figure-4: Collection efficiency vs number N of 5nm-Ge layers in the intrinsic region of a GaAs p-i-n cell. Note** 

**Figure 4.** Collection efficiency vs number N of 5nm-Ge layers in the intrinsic region of a GaAs p-i-n cell. Note that 42%

Note from Figure 4, that 42% collection efficiency can be expected at 400 suns and at large number of superlattice periods (N ~ 200). By using (11) in (17) we propose the total efficiency as the sum of two conclude by re-writing as the sum of two factors bulk efficiency and excess

Note from Figure 4, that 42% collection efficiency can be expected at 400 suns and at large number of superlattice periods (N ~ 200). By using (11) in (17) we propose the total efficiency as the sum of two conclude by re-writing as the sum of two factors bulk efficiency and excess

> *in oc*

(17)

1/ 2 '

N-period superlattice, under concentrated power equivalent to X suns. We would also like to see

*<sup>N</sup> V VX*

= += + + (19)

*NX V FF P*

1/2 '

*in*


of a cell, enriched with an embedded

*NX <sup>V</sup> FF*

Expression (17) essentially predicts excess efficiency *δη* of a cell, enriched with an embedded N-period superlattice, under concentrated power equivalent to X suns. We would also like to

(%) (1.557 ) *oc*

**that 42% is feasible with 200 periods at 400 suns (XPin = 400mW/cm^2).** 

*bulk bulk P*

*bulk bulk*

Expression (17) essentially predicts excess efficiency

see excess efficiency against X at a fixed N: (17) can be re-written:

*in*

excess efficiency against X at a fixed N: (17) can be re-written:

( ln )) 1.323( ) 26.26 *<sup>V</sup> <sup>V</sup> <sup>X</sup> P X N*

*oc t*

1.323( ) 26.26 ( ln )) *bulk oc t in*

*P X*

Note however from (18) that the excess efficiency δη strongly depends on X-1/2 and N. High X

Note from Figure-5 that high X does not necessarily mean better performance; at 100 and 400 suns, 30 SL layers in the intrinsic region will provide 30.88% and 28.63% collection respectively;

(18)

(%) (1.557 )

hhdh

"slows down" excess efficiency as shown from Figure-5:

hhdhh

is feasible with 200 periods at 400 suns (XPin=400mW/cm^2).

efficiency:

190 Solar Cells - New Approaches and Reviews

 

efficiency:

 *bulk*  this is because excess efficiency gains (δη) drop quickly with X (Fig. 5 and expression (18)); high concentration does not increase the second term of (17) or (18). For instance, compromis‐ ing with N=50 periods, 100 suns would suffice for excess efficiency δη=7.7% (or total efficiency of 25.50+7.7=33.20% according to (18). As the concentration increases, the excess term decreases down to the bulk cell's efficiency. On the other hand, as long as the concentration is kept between 100 and 500 suns, considerable increase in efficiency is plausible as seen in Figure 5. A summary of cell performance improvement is shown below (Table 1) where (i) bulk cell at one sun (ii) bulk cell at 400 suns and (iii) hybrid cell at 400 suns is depicted. Note from Figure-5 that high X does not necessarily mean better performance; at 100 and 400 suns, 30 SL layers in the intrinsic region will provide 30.88% and 28.63% collection respectively; this is because excess efficiency gains ( drop quickly with X (Fig. 5 and expression (18)); high concentration does not increase the second term of (17) or (18). For instance, compromising with N = 50 periods, 100 suns would suffice for excess efficiency=

Table-1 summarizes some results predicted at 1 and 400 suns respectively; note (a) ideal efficiency at 400 suns with varying period number N. 7.7% (or total efficiency of 25.50 + 7.7 = 33.20% according to (18). As the concentration increases, the excess term decreases down to the bulk cell's efficiency. On the other hand, as

long as the concentration is kept between 100 and 500 suns, considerable increase in efficiency is



**Table 1.** Bulk and hybrid cell at 1 and 400 suns (Pin=100mW/cm2 , X=1); The bulk p-i-n GaAs cell is simulated as a standard 1-cm2 PV device.

Notice improvements in the efficiency of the bulk pin GaAs cell: from 25.42% (X=1) to 26.25% (X=400) and 28.63% at X=400, N=30, and 30.21% at X=400, N=50 (compare with Fig-16 of reference [16] good agreement in overall efficiency for p(GaAs)-I(quantum well)-n(GaAs) strained-layer cell).


Table-2 below summarizes improvements on cell performance:

[α]: (25.73) (Voc+Vt lnX) (0.85)=26.26%; [b]: N=50.

(Where *δ* = 1.323*VOC* ' *Pin <sup>X</sup>* (see (18)), and VOC (bulk, one sun)=1.051V).

**Table 2.** Comparison between bulk and hybrid cell at various solar concentrations

Note that 50% efficiency is feasible by posing the question: what is the N value in the intrinsic region of the host cell at a given X? Table 3 below summarizes device design for high per‐ formance; for instance, 50% efficiency is feasible with a 1.5 μm superlattice at 400 suns. Table 3 summarizes and compares bulk and hybrid cell performance at different concentration levels. For instance, note that 50% efficiency is feasible under 300 suns with 260 Ge layers. Of course, cost reduction and high efficiency levels would lead to the obvious choice (Table-3)


with 100 suns illuminating a 160-period superlattice in a p-i-n GaAs cell with 10 um-long intrinsic region.

**Cell properties & conditions**

192 Solar Cells - New Approaches and Reviews

standard 1-cm2

strained-layer cell).

**Hybrid cell OCvoltage V'=Voc+Vt**

[α]: (25.73) (Voc+Vt

1.323*VOC* '

(Where *δ* =

**lnX**

**GaAs p-i-n cell**

η (%) 25.42 26.26

Table-2 below summarizes improvements on cell performance:

**X(suns) δ/Ν Excessefficiency**

**Table 1.** Bulk and hybrid cell at 1 and 400 suns (Pin=100mW/cm2

lnX) (0.85)=26.26%; [b]: N=50.

*Pin <sup>X</sup>* (see (18)), and VOC (bulk, one sun)=1.051V).

**Table 2.** Comparison between bulk and hybrid cell at various solar concentrations

PV device.

X (suns) 1 400 400

Notice improvements in the efficiency of the bulk pin GaAs cell: from 25.42% (X=1) to 26.25% (X=400) and 28.63% at X=400, N=30, and 30.21% at X=400, N=50 (compare with Fig-16 of reference [16] good agreement in overall efficiency for p(GaAs)-I(quantum well)-n(GaAs)

1.166 100 0.154 7.70 25.50 33.20 1.183 200 0.111 5.55 25.87 31.42 1.194 300 0.091 4.55 26.11 30.66 1.201 400 0.079 3.95 26.26[a] 30.21 1.206 500 0.071 3.55 26.37 29.92

Note that 50% efficiency is feasible by posing the question: what is the N value in the intrinsic region of the host cell at a given X? Table 3 below summarizes device design for high per‐ formance; for instance, 50% efficiency is feasible with a 1.5 μm superlattice at 400 suns. Table 3 summarizes and compares bulk and hybrid cell performance at different concentration levels. For instance, note that 50% efficiency is feasible under 300 suns with 260 Ge layers. Of course, cost reduction and high efficiency levels would lead to the obvious choice (Table-3)

**δη (%)**

**GaAs p-i-n cell (no SL)** **Hybrid cell with GaAs/Ge superlattice (SL)**

> 28.63 (N =30) 30.21 (N= 50)[16] 32.58 (N = 80) 38.11 (N 150) 42.06 (N = 200) 52.33 (N = 330)

, X=1); The bulk p-i-n GaAs cell is simulated as a

**Bulk cell Efficiency,**

**Hybrid cell Efficiency [b]**

Table-3 above is an extrapolation to higher efficiencies, feasible through the proposed structure (Figure 6). We set a limit at 50% and simulate the feasibility of the structure proposed. The latter is a p-i-n cell with a superlattice in the middle of the intrinsic region; the superlattice is **Table 3.** Choice of number N periods for 50% total ideal efficiency at different solar concentrations X; the last column indicates the length of the SL region (length of intrinsic region Li =10μm); last column compares superlattice vs intrinsic region widths. We set a lowest limit of X=100 to ensure negligible dark carrier concentration (see also expression (7)). High X increases current but not necessarily the efficiency.

illuminated at X-suns and produces excess thermionic current density from two-dimensional systems (quantum wells). The most reasonable choice is indicated by the first row of Table-3, where 50% can be reached (X=100, N=160; =50%). Such designs and high efficiency options are perfectly suited for concentration photovoltaics (CPV); note also that maturity of (a) current light-concentrating systems and (b) device enrichment via routine MBE growth-techniques make the proposed design reasonable for production. On the other hand, three junction cells (MJ) are equally complicated structures because they include three cells in series with highly doped AlGaAs/GaAs tunnel-junctions linking them. Our proposed cell is a bulk GaAs cell enriched with a matched N-period superlattice and without the need of any tunnel junction. The proposed device relies on the bulk properties of a GaAs cell which in turn is enriched with an implanted superlattice strip that provides excess current under X > 100. Its twofold-advantage over current multi-junction (MJ) cells is (a) absence of top cell (hence no photon shadowing effects) and (b) absence of tunnel junctions (TJ). Such a superlattice grown in the mid-intrinsic region of an all-GaAs p-i-n solar cell illuminated at X suns is depicted in Figure-6 below. Light is supposed to be focused on the interior of the device covering the superlattice where 1eV photons are expected to be strongly absorbed and from where photo-carriers are expected to thermionically escape. Table-3 above is an extrapolation to higher efficiencies, feasible through the proposed structure (Figure 6). We set a limit at 50% and simulate the feasibility of the structure proposed. The latter is a p-i-n cell with a superlattice in the middle of the intrinsic region; the superlattice is illuminated at X-suns and produces excess thermionic current density from two-dimensional systems (quantum wells). The most reasonable choice is indicated by the first row of Table-3, where 50% can be reached (X=100, N=160; η=50%). Such designs and high efficiency options are perfectly suited for concentration photovoltaics (CPV); note also that maturity of (a) current light-concentrating systems and (b) device enrichment via routine MBE growth-techniques make the proposed design reasonable for production. On the other hand, three junction cells (MJ) are equally complicated structures because they include three cells in series with highly doped AlGaAs/GaAs tunnel-junctions linking them. Our proposed cell is a bulk GaAs cell enriched with a matched N-period superlattice and without the need of any tunnel junction. The proposed device relies on the bulk properties of a GaAs cell which in turn is enriched with an implanted superlattice strip that provides excess current under X > 100. Its twofoldadvantage over current multi-junction (MJ) cells is (a) absence of top cell (hence no photon shadowing effects) and (b) absence of tunnel junctions (TJ). Such a superlattice grown in the mid-intrinsic region of an all-GaAs p-i-n solar cell illuminated at X suns is depicted in Figure-6 below. Light is supposed to be focused on the interior of the device covering the superlattice where 1eV photons are expected to be strongly absorbed and from where photo-carriers are expected to thermionically escape.

**matched superlattice (SL) in its intrinsic region. Total efficiency of the hybrid cell is the sum of bulk cell efficiency and excess efficiency. Implanted SLs in the bulk are feasible based on routine MBE techniques [15]. Details of the optical system and the spot covering most of the SL region are under study. Figure 6.** Concentrated solar radiation incident on a 1-cm2 GaAs p-i-n cell with a GaAs/Ge lattice-matched superlattice (SL) in its intrinsic region. Total efficiency of the hybrid cell is the sum of bulk cell efficiency η and excess efficiency. Implanted SLs in the bulk are feasible based on routine MBE techniques [15]. Details of the optical system and the spot covering most of the SL region are under study.

 **GaAs p-i-n cell with a GaAs/Ge lattice-**

**Figure-6: Concentrated solar radiation incident on a 1-cm<sup>2</sup>**

As seen from Figure 6, a superlattice can become the source for excess carriers in the conduction band if light can be concentrated on it inside the pin cell. This is an immediate advantage: the SL As seen from Figure 6, a superlattice can become the source for excess carriers in the conduction band if light can be concentrated on it inside the pin cell. This is an immediate advantage: the SL region generates excess carriers and causes efficiency increase δη as seen in expression (18):

region generates excess carriers and causes efficiency increase as seen in expression (18): ( ln ) 1.323( ) *<sup>V</sup> <sup>V</sup> <sup>X</sup> N δη* <sup>=</sup> 1.323(*<sup>N</sup>* ) *Pin <sup>X</sup>* (*Voc* <sup>+</sup> *Vt*ln*<sup>X</sup>* ); (No losses due to excess carrier scattering are assumed, see also [12])

*P X oc t in* ; (No losses due to excess carrier scattering are assumed, see also [12]) Figure-6 presents the main concept: a lattice-matched superlattice in the middle of the bulk GaAs i-layer of the control cell and illuminated at X suns improves cell performance through thermionic escape of photo-electrons from individual quantum wells. In our analysis we assume 100% escape rate of these electrons from their quantum traps [12] and an ideal device performance in terms of quality factors and shunt resistance.
