**2. Theoretical study on optimization of high efficiency multi-junction solar cells**

In designing GaInP/GaInAs/Ge triple-junction cells, the principles for maximising cell effi‐ ciency are: (1) increasing the amount of light collected by each cell that is turned into carri‐ ers, (2) increasing the collection of light-generated carriers by each p-n junction, (3) minimising the forward bias dark current, and (4) photocurrents matching among sub-cells.

In practice, basic designs for these solar cells involve various doping concentrations and lay‐ er thicknesses for the window, emitter, base, and back surface field (BSF) regions in each sub-cell. In order to optimize the designs, a rigorous model including optical and electrical modules was developed to analyse the bulk parameters effect on the external quantum effi‐ ciency, photocurrent and photovoltage of the GaInP/GaInAs/Ge multi-junction solar cells.

#### **2.1. Theoretical approach**

tionally, these devices are the only solar cells currently available with efficiencies above 30%. The high efficiency is due to the reduction of thermalization and transmission losses in

Future terrestrial cells will likely feature four or more junctions with a performance poten‐ tial capable of reaching over 45% efficiency at concentration of hundreds of suns. The 4-, 5-, or 6-junction solar cells with concentrator trade lower current densities for higher voltage and divide the solar spectrum more efficiently. The lower current densities in these cells can

High-efficiency GaInP/GaAs/InGaAs triple-junction solar cells grown inversely with a meta‐ morphic bottom junction could be achieved by replacing the bottom Ge sub-cell with 1 eV en‐ ergy gap material. In0.3Ga0.7As is the promised candidate, if without the lattices mis-match (LMM, around 2%) with the other two sub-cells. Therefore, the LM top and middle sub-cells were grown first, and the graded buffers were employed between middle and bottom cells to overcome the mismatch and to prevent the threading dislocations. The substrate was removed for the reusing. This inverted metamorphic, monolithic triple junction solar cells could be ob‐

A metamorphic Ga0.35In0.65P/Ga0.83In0.17As/Ge triple-junction solar cell is studied to provide current-matching of all three sub-cells and thus give a device structure with virtually ideal energy gap combination. It is demonstrated that the key for the realization of this device is the improvement of material quality of the lattice-mismatched layers as well as the develop‐ ment of a highly relaxed Ga1−yInyAs buffer structure between the Ge substrate and the mid‐ dle cell. This allows the metamorphic growth with low dislocation densities below 106 cm-2. The performance of the device has been demonstrated by a conversion efficiency of 41.1% at

In this chapter, the theoretical and experimental investigation of the most sophisticated, in‐ dustrialized and commercialized GaInP/GaInAs/Ge triple junction solar cell was extensively described. Accelerated aging tests of the high concentration multi-junction solar cells and

**2. Theoretical study on optimization of high efficiency multi-junction**

In designing GaInP/GaInAs/Ge triple-junction cells, the principles for maximising cell effi‐ ciency are: (1) increasing the amount of light collected by each cell that is turned into carri‐ ers, (2) increasing the collection of light-generated carriers by each p-n junction, (3) minimising the forward bias dark current, and (4) photocurrents matching among sub-cells. In practice, basic designs for these solar cells involve various doping concentrations and lay‐ er thicknesses for the window, emitter, base, and back surface field (BSF) regions in each sub-cell. In order to optimize the designs, a rigorous model including optical and electrical

tained with at least 2% higher efficiency than the traditional one theoretically [7].

discussions on outdoor power plant performances were also presented.

R) at high concentrations of suns when com‐

solar cells when the number of p-n junctions is increased.

significantly reduce the resistive power loss (I2

pared with the 3-junction cell [6].

444 Optoelectronics - Advanced Materials and Devices

454 suns AM1.5D [8].

**solar cells**

We present here a brief description of the equations used in our model. Thorough treat‐ ments of photovoltaic devices can be found elsewhere [9]. A schematic of a typical latticematched GaInP/GaInAs/Ge solar cell is shown in Figure1. It consists of an n/p GaInP junction on top of an n/p GaInAs junction which lies on an n/p Ge junction. The triple junc‐ tion cells are series connected by two p++/n++ tunnel junctions.

**Figure 1.** Solar cell structure used for simulation.

The solar spectrum, striking the front of the cell, includes ultraviolet, visible, and infrared lights. The absorption coefficient for short-wavelength light is quite large, and most of the blue light is absorbed very close to the front of the cell for generating photo carriers. Light with energy slightly larger than the energy gap is weakly absorbed throughout the cell. Light with energy less than the energy gap passes through the front cell and is absorbed in the next one. The photo carriers generated by the short-wavelength light diffuse inside the cell until they are either collected at the p-n junction or recombined with a majority carrier in bulk or at interface. The efficiency of the solar cell increase when all the photo carriers are collected at the junction instead of recombining elsewhere. Thus, recombination is at the front and back of the cell effects on the efficiency of the cell.

At the first level approximation, multi-junction cells behave like homo-junction cells in ser‐ ies, so their open circuit voltage is the summation of the voltages of the sub-cells, while their short circuit current is that of the sub-cell with the smallest current. Hence, the performance of a multi-junction cell can be obtained from the performance of each sub-cell, evaluated in‐ dependently. The load current density *J* is represented by the superposition of two diode currents and the photo-generated current,

$$J = J\_{ph} - J\_{01}(e^{qV/kT} - 1) - J\_{02}(e^{qV/2kT} - 1) \tag{1}$$

*J*01,*emitter* =*q*

*J*01,*base* =*q*

ness *W* , are given by [10],

*ni* 2 *ND*

*ni* 2 *NA* *Dp L <sup>p</sup>*

*Dn L <sup>n</sup>*

and *R* is the reflectance of the anti-reflective coating. *ni*

{ *SpL <sup>p</sup>* / *Dp*cosh (*de* <sup>−</sup>*Wn*)/ *<sup>L</sup> <sup>p</sup>*

*SpL <sup>p</sup>* / *Dp*sinh (*de* −*Wn*) / *L <sup>p</sup>*

*<sup>J</sup>*<sup>02</sup> <sup>=</sup> *<sup>W</sup> ni*

transmission of incident photon flux into the sub-cell under consideration.

*Vd* =*kT* log(

*ND* + *NA NDNA*

whereas *Dp* , *Dn* , *L <sup>p</sup>* , *L <sup>n</sup>* and *τ* depend on the doping concentration [11].

*W* = 2*ε*

Where *q* is electron charge, *F* the incident photon flux, *α* is an optical absorption coefficient

*NA* and *ND* are the concentrations of acceptors and donors. *de* is the emitter thickness, *db* the base thickness, *L <sup>p</sup>* the hole diffusion length in the emitter, *L <sup>n</sup>* the electron diffusion length in the base, *Sp* the hole surface recombination velocity in the emitter, *Sn* the electron surface recombination velocity in the base, *Dp* the hole diffusion coefficient in the emitter, *Dn* the electron diffusion coefficient in the base, and *τ* is the non-radiative carrier lifetime. *TF* is the

The build-in voltage *Vd* of the junction, the thickness of the depleted layer in the emitter *Wn* , the thickness of the depleted layer in the base *Wp* , and the total depleted zone thick‐

> *NDNA ni*

Where *k* is the Boltzmann constant, *ε* the dielectric constant and *T* the temperature ( *T* =25 °C was used in this paper). It is important to note that *F* and *α* depend on the wavelength,

The optical model proposed in this paper is based on the transfer matrix formalism. It al‐ lows calculating the incident optical spectrum on each sub-cell from the solar spectrum. Each layer of the multi-junction is described by a transfer matrix *M* which is defined by

{ *SnL <sup>n</sup>* / *Dn*cosh (*db* <sup>−</sup>*Wp*) / *<sup>L</sup> <sup>n</sup>* <sup>+</sup> sinh (*db* <sup>−</sup>*Wp*) / *<sup>L</sup> <sup>n</sup> SnL <sup>n</sup>* / *Dn*sinh (*db* −*Wp*)/ *L <sup>n</sup>* + cosh (*db* −*Wp*) / *L <sup>n</sup>*

+ sinh (*de* −*Wn*) / *L <sup>p</sup>*

} (7)

447

III-V Multi-Junction Solar Cells http://dx.doi.org/10.5772/50965

} (8)

+ cosh (*de* −*Wn*) / *L <sup>p</sup>*

2(*Vd* <sup>−</sup>*<sup>V</sup>* )*<sup>τ</sup>* (9)

is the intrinsic carrier concentration,

<sup>2</sup> ) (10)

(*Vd* <sup>−</sup>*<sup>V</sup>* <sup>−</sup>2*kT* ) (11)

*Wn* =*W* / (1 + *ND* / *NA*) (12)

*Wp* =*W* −*Wn* (13)

Where *J ph* is the photocurrent density, *J*01 the ideal dark saturation current component and *J*02 the space charge non-ideal dark saturation current component.

The photocurrent density and dark current density are given by the sum of the photocur‐ rents and the sum of the dark current density, respectively, generated in the emitter, the base and the depleted region of the cell. [9] We have

$$J\_{\
uph} = J\_{\
u
u
u
itter} + J\_{\
uuse} + J\_{\
dejected} \tag{2}$$

$$\begin{aligned} \left| J\_{emitter} = \frac{qF(1-R)\alpha L\_p}{\left(\alpha L\_p\right)^2 - 1} \right| \\ \left| \frac{S\_p L\_p}{D\_p} + \alpha L\_p - e^{-\alpha \left(d\_e - W\_n\right)} \right| \\ \left| \frac{S\_p L\_p}{D\_p} \cosh\left[\left(d\_e - W\_n\right)/L\_p\right] + \sinh\left[\left(d\_e - W\_n\right)/L\_p\right]} \right| \\ \left| \frac{S\_p L\_p}{D\_p} \sinh\left[\left(d\_e - W\_n\right)/L\_p\right] + \cosh\left[\left(d\_e - W\_n\right)/L\_p\right]} \right| \\ - \alpha L\_p e^{-\alpha \left(d\_e - W\_n\right)} \end{aligned} \tag{3}$$

$$\begin{aligned} \left\| \begin{array}{l} J\_{\text{base}} = \frac{qF\left(1-R\right)\alpha L\_{\text{n}}}{\left(\alpha L\_{\text{n}}\right)^{2} - 1} e^{-\alpha\left(d\_{b} - W\_{\text{n}} + W\right)}\\ \vdots\\ \frac{S\_{n}L\_{\text{n}}}{D\_{n}} \Big(\cosh\llbracket\left(d\_{c} - W\_{\text{n}}\right)/L\_{\text{n}}\right) - e^{-\alpha\left(d\_{b} - W\_{\text{p}}\right)}\\ \alpha L\_{\text{n}} - \frac{+\sinh\llbracket\left(d\_{b} - W\_{\text{p}}\right)/L\_{\text{n}}\right] + \alpha L\_{\text{n}}e^{-\alpha\left(d\_{b} - W\_{\text{p}}\right)}\\ \frac{S\_{n}L\_{\text{n}}}{D\_{n}}\text{sinh}\llbracket\left(d\_{b} - W\_{\text{p}}\right)/L\_{\text{n}}\Bigr] + \cosh\llbracket\left(d\_{b} - W\_{\text{p}}\right)/L\_{\text{n}}\Bigr\end{array} \tag{4}$$

$$J\_{\
u applied} = qF \left( 1 - R \right) e^{-\alpha \left( d\_e - \mathcal{W}\_n \right)} \left( 1 - e^{-\alpha \mathcal{W}} \right) \tag{5}$$

$$J\_{01} = J\_{01, 
ewitter} + J\_{01, 
base} \tag{6}$$

$$J\_{01,emitter} = q \frac{n\_i^2}{N\_D} \frac{D\_p}{L\_p} \left| \frac{S\_p L\_p / D\_p \cosh[\left(d\_\varepsilon - W\_n\right)/L\_p]}{S\_p L\_p / D\_p \sinh[\left(d\_\varepsilon - W\_n\right)/L\_p]} + \cosh[\left(d\_\varepsilon - W\_n\right)/L\_p]} \right| \tag{7}$$

short circuit current is that of the sub-cell with the smallest current. Hence, the performance of a multi-junction cell can be obtained from the performance of each sub-cell, evaluated in‐ dependently. The load current density *J* is represented by the superposition of two diode

Where *J ph* is the photocurrent density, *J*01 the ideal dark saturation current component and

The photocurrent density and dark current density are given by the sum of the photocur‐ rents and the sum of the dark current density, respectively, generated in the emitter, the

*J*02 the space charge non-ideal dark saturation current component.

*qF* (1−*R*)*αL <sup>p</sup>* (*αL <sup>p</sup>*)<sup>2</sup> −1

−*α*(*de*−*Wn*)

−*α*(*db*−*Wn*+*W* )

+sinh (*db* −*Wp*) / *L <sup>n</sup>* + *αL <sup>n</sup>e*

(cosh (*de* −*Wn*) / *L <sup>n</sup>* −*e*

sinh (*db* −*Wp*) / *L <sup>n</sup>* + cosh (*db* −*Wp*)/ *L <sup>n</sup>*

−*α*(*de*−*Wn*)

cosh (*de* −*Wn*) / *L <sup>p</sup>*

sinh (*de* −*Wn*) / *L <sup>p</sup>*

+ *αL <sup>p</sup>* −*e*

−*α*(*de*−*Wn*)

*SnL <sup>n</sup> Dn*

*Jdepleted* =*qF* (1−*R*)*e*

*qF* (1−*R*)*αL <sup>n</sup>* (*α<sup>L</sup> <sup>n</sup>*)<sup>2</sup> <sup>−</sup><sup>1</sup> *<sup>e</sup>*

> *SnL <sup>n</sup> Dn*

base and the depleted region of the cell. [9] We have

*Jemitter* =

{ *SpL p Dp*

( *SpL <sup>p</sup> Dp*

> *SpL <sup>p</sup> Dp*

−*αL <sup>p</sup>e*

*Jbase* =

{*α<sup>L</sup> <sup>n</sup>* <sup>−</sup>

*<sup>J</sup>* <sup>=</sup> *<sup>J</sup> ph* <sup>−</sup> *<sup>J</sup>*01(*<sup>e</sup> qV* /*kT* <sup>−</sup>1)<sup>−</sup> *<sup>J</sup>*02(*<sup>e</sup> qV* /2*kT* <sup>−</sup>1) (1)

*J ph* = *Jemitter* + *Jbase* + *Jdepleted* (2)

} (3)

} (4)

(1−*e* <sup>−</sup>*α<sup>W</sup>* ) (5)

+ sinh (*de* −*Wn*) / *L <sup>p</sup>* )

+ cosh (*de* −*Wn*) / *L <sup>p</sup>*

−*α*(*db*−*Wp*) )

−*α*(*db*−*Wp*)

*J*<sup>01</sup> = *J*01,*emitter* + *J*01,*base* (6)

currents and the photo-generated current,

446 Optoelectronics - Advanced Materials and Devices

$$J\_{01, \text{base}} = q \frac{n\_i^2}{N\_A} \frac{D\_n}{L\_n} \left| \frac{S\_n L\_n / D\_n \cosh[\left(d\_b - \mathcal{W}\_p\right)/L\_n] + \sinh[\left(d\_b - \mathcal{W}\_p\right)/L\_n]}{S\_n L\_n / D\_n \sinh[\left(d\_b - \mathcal{W}\_p\right)/L\_n] + \cosh[\left(d\_b - \mathcal{W}\_p\right)/L\_n]} \right| \tag{8}$$

$$J\_{o2} = \frac{W n\_i}{2(V\_d - V)\tau} \tag{9}$$

Where *q* is electron charge, *F* the incident photon flux, *α* is an optical absorption coefficient and *R* is the reflectance of the anti-reflective coating. *ni* is the intrinsic carrier concentration, *NA* and *ND* are the concentrations of acceptors and donors. *de* is the emitter thickness, *db* the base thickness, *L <sup>p</sup>* the hole diffusion length in the emitter, *L <sup>n</sup>* the electron diffusion length in the base, *Sp* the hole surface recombination velocity in the emitter, *Sn* the electron surface recombination velocity in the base, *Dp* the hole diffusion coefficient in the emitter, *Dn* the electron diffusion coefficient in the base, and *τ* is the non-radiative carrier lifetime. *TF* is the transmission of incident photon flux into the sub-cell under consideration.

The build-in voltage *Vd* of the junction, the thickness of the depleted layer in the emitter *Wn* , the thickness of the depleted layer in the base *Wp* , and the total depleted zone thick‐ ness *W* , are given by [10],

$$V\_d = kT \log(\frac{N\_D N\_A}{n\_i^2}) \tag{10}$$

$$W = \sqrt{2\varepsilon \frac{N\_D + N\_A}{N\_D N\_A} (V\_d - V - 2kT)}\tag{11}$$

$$\mathcal{W}\_n = \mathcal{W} \;/\left(1 + \mathcal{N}\_D / \mathcal{N}\_A\right) \tag{12}$$

$$\text{IV}\_p = \text{IV} - \text{IV}\_n \tag{13}$$

Where *k* is the Boltzmann constant, *ε* the dielectric constant and *T* the temperature ( *T* =25 °C was used in this paper). It is important to note that *F* and *α* depend on the wavelength, whereas *Dp* , *Dn* , *L <sup>p</sup>* , *L <sup>n</sup>* and *τ* depend on the doping concentration [11].

The optical model proposed in this paper is based on the transfer matrix formalism. It al‐ lows calculating the incident optical spectrum on each sub-cell from the solar spectrum. Each layer of the multi-junction is described by a transfer matrix *M* which is defined by

$$\begin{aligned} \, \_{0,M} = \begin{vmatrix} M\_{0,0} M\_{0,1} \\ M\_{1,0} M\_{1,1} \end{vmatrix} = \left| \cos \left( d \frac{2 \pi \left( n - i \lambda \alpha / 4 \pi \right)}{\lambda} \right) i \frac{\sin \left( d \frac{2 \pi \left( n - i \lambda \alpha / 4 \pi \right)}{\lambda} \right)}{\left( n - i \lambda \alpha / 4 \pi \right)} \right| \\ \times \left| n - i \lambda \alpha / 4 \pi \right| \sin \left| d \frac{2 \pi \left( n - i \lambda \alpha / 4 \pi \right)}{\lambda} \right| \cos \left( d \frac{2 \pi \left( n - i \lambda \alpha / 4 \pi \right)}{\lambda} \right) \end{aligned} \tag{14}$$

Where *n* and *d* are the refraction index and the thickness of the layer, respectively. The transmission coefficient *TM* [12] of the layer is then given by

$$T\_M = \frac{4n\_0^2}{(n\_0M\_{0,0} + n\_0n\_sM\_{0,1} + M\_{1,0} + n\_sM\_{1,1})^2} \tag{15}$$

Where *n*0 is the superstrate refraction index and *ns* is the substrate refraction index of the sub-cell. The *Mi*, *<sup>j</sup>* coefficients refer to the matrix transfer elements. Thus, it is possible to de‐ termine the incident spectrum on each sub-cell. The incident photon flux in GaInP, GaInAs and Ge sub-cells are given by

$$F\_{GalnP} = T\_{ARC} F\_{solar} \tag{16}$$

**Parameter Ge GaInAs GaInP** *Dn*(cm2/s) 22.86 140.02 29.39 *Dp*(cm2/s) 10.71 4.02 1.03 *L <sup>n</sup>*(cm) 5.3х10-3 9.7х10-4 6.3х10-4 *L <sup>p</sup>*(cm) 8.8х10-4 7.3х10-5 3.7х10-5 τ (s) 8.9х10-7 8.9х10-9 4.2х10-9

to be about 0.52 micrometers with doping concentration of about 1x1017 cm-3.

The direct gap absorption spectra of the bulk Ge was used for calculation

*αGe* =1.9 (*E* −*Eg*

Where *E* is the photon energy and *Eg* the fundamental energy gap, both in eV, and *α* in 1/

The diffusion length, the diffusion coefficient and the nonradiative carrier lifetime are calcu‐ lated as a function of the doping concentration. The material parameters used for calculation

The absorption coefficient of the GaInAs (with In content of about 0.01) can be fitted by

The absorption coefficient of the GaInP can be fitted by

micrometers.

are summarized in table 1.

As shown in Figure 1, typical two-terminal triple-junction cells for space application with a Ge bottom cell, a GaInAs middle cell and a GaInP top cell with energy gaps of 0.661, 1.405 and 1.85 eV, respectively. The Ge cell is built on the p-type initial substrate; therefore, the Ge base is about 150 micrometers thick, with doping concentration of about 6x1017 cm-3; the Ge emitter is about 0.3 micrometers thick, with an n-type doping concentration of about 1x1019 cm-3. The emitters for the other two cells are 0.1 micrometers thick with doping concentra‐ tion of about 1x1018 cm-3. Since the AM0 spectrum contains relatively more high-energy pho‐ tons with energy greater than the GaInP top cell's energy gap, triple-junction cell with a very thick top cell will generally be photocurrent limited by the middle (GaInAs) cell. Therefore, the middle cell thickness was set to be thick enough (3.6 micrometers in this paper) with the doping concentration of about 2x1017 cm-3, and the optimal top cell thickness was suggested

*αGaInP* =5.5 (*E* −*Eg*) + 1.5 (*E* − *Eg* −1) (19)

*αGaInA*s=3.3 (*E* −*Eg*) (20)

*<sup>Γ</sup>*)/ *E* (21)

III-V Multi-Junction Solar Cells http://dx.doi.org/10.5772/50965 449

**Table 1.** Material parameters used for calculation in this paper.

$$F\_{GalnAs} = T\_{ARC} T\_{GalnP} F\_{solar} \tag{17}$$

$$F\_{Ge} = T\_{GalnAs} T\_{GalnP} T\_{ARC} F\_{solar} \tag{18}$$

where *Fsolar* is the incident photon flux, *T ARC* , *TGaInP* and *TGaInAs* are the transmission coeffi‐ cient of the anti-reflective coating, the GaInP sub-cell and the GaInAs sub-cell, respectively. This model includes optical and electrical modules. Thus, it allows the calculation of the quantity of photons arriving at each junction from the solar spectrum. Then, the electrical model calculates, via interface recombination velocity, the photocurrents in the space charge region, the emitter and the base for each junction.

#### **2.2. Solar cell structures and parameters**

To calculate the power production of the GaInP/GaInAs/Ge triple-junction cells for space applications, the incident photon flux *Fsolar* was taken from a newly proposed reference air mass zero (AM0) spectra (ASTM E-490). The integration of ASTM E490 AM0 solar spectral irradiance has been made to conform to the value of the solar constant accepted by the space community, which is 1366.1 W/m2 . The transmission coefficient of the antireflective coating *T ARC* was set to be a constant of 98%, while the transmission coeffi‐ cients of the GaInP sub-cell and the GaInAs sub-cell are calculated according to Eqs.14 and 15, which have wavelength dependence.


**Table 1.** Material parameters used for calculation in this paper.

*M* =(

*M*0,0*M*0,1 *M*1,0*M*1,1

448 Optoelectronics - Advanced Materials and Devices

and Ge sub-cells are given by

region, the emitter and the base for each junction.

by the space community, which is 1366.1 W/m2

and 15, which have wavelength dependence.

**2.2. Solar cell structures and parameters**

) =(cos(*<sup>d</sup>* <sup>2</sup>*π*(*<sup>n</sup>* <sup>−</sup>*iλα* / <sup>4</sup>*π*)

transmission coefficient *TM* [12] of the layer is then given by

*TM* <sup>=</sup> <sup>4</sup>*n*<sup>0</sup>

*<sup>λ</sup>* )*<sup>i</sup>*

Where *n* and *d* are the refraction index and the thickness of the layer, respectively. The

2

Where *n*0 is the superstrate refraction index and *ns* is the substrate refraction index of the sub-cell. The *Mi*, *<sup>j</sup>* coefficients refer to the matrix transfer elements. Thus, it is possible to de‐ termine the incident spectrum on each sub-cell. The incident photon flux in GaInP, GaInAs

where *Fsolar* is the incident photon flux, *T ARC* , *TGaInP* and *TGaInAs* are the transmission coeffi‐ cient of the anti-reflective coating, the GaInP sub-cell and the GaInAs sub-cell, respectively. This model includes optical and electrical modules. Thus, it allows the calculation of the quantity of photons arriving at each junction from the solar spectrum. Then, the electrical model calculates, via interface recombination velocity, the photocurrents in the space charge

To calculate the power production of the GaInP/GaInAs/Ge triple-junction cells for space applications, the incident photon flux *Fsolar* was taken from a newly proposed reference air mass zero (AM0) spectra (ASTM E-490). The integration of ASTM E490 AM0 solar spectral irradiance has been made to conform to the value of the solar constant accepted

reflective coating *T ARC* was set to be a constant of 98%, while the transmission coeffi‐ cients of the GaInP sub-cell and the GaInAs sub-cell are calculated according to Eqs.14

(*<sup>n</sup>* <sup>−</sup>*iλα* / <sup>4</sup>*π*)sin(*<sup>d</sup>* <sup>2</sup>*π*(*<sup>n</sup>* <sup>−</sup>*iλα* / <sup>4</sup>*π*)

sin(*<sup>d</sup>* <sup>2</sup>*π*(*<sup>n</sup>* <sup>−</sup>*iλα* / <sup>4</sup>*π*)

(*n* −*iλα* / 4*π*)

*<sup>λ</sup>* )

*<sup>λ</sup>* )cos(*<sup>d</sup>* <sup>2</sup>*π*(*<sup>n</sup>* <sup>−</sup>*iλα* / <sup>4</sup>*π*)

(*n*0*M*0,0 <sup>+</sup> *<sup>n</sup>*0*nsM*0,1 <sup>+</sup> *<sup>M</sup>*1,0 <sup>+</sup> *nsM*1,1)<sup>2</sup> (15)

*FGaInP* =*T ARCFsolar* (16)

*FGaInAs* =*T ARCTGaInP Fsolar* (17)

*FGe* =*TGaInAsTGaInPT ARCFsolar* (18)

. The transmission coefficient of the anti-

*<sup>λ</sup>* )

) (14)

As shown in Figure 1, typical two-terminal triple-junction cells for space application with a Ge bottom cell, a GaInAs middle cell and a GaInP top cell with energy gaps of 0.661, 1.405 and 1.85 eV, respectively. The Ge cell is built on the p-type initial substrate; therefore, the Ge base is about 150 micrometers thick, with doping concentration of about 6x1017 cm-3; the Ge emitter is about 0.3 micrometers thick, with an n-type doping concentration of about 1x1019 cm-3. The emitters for the other two cells are 0.1 micrometers thick with doping concentra‐ tion of about 1x1018 cm-3. Since the AM0 spectrum contains relatively more high-energy pho‐ tons with energy greater than the GaInP top cell's energy gap, triple-junction cell with a very thick top cell will generally be photocurrent limited by the middle (GaInAs) cell. Therefore, the middle cell thickness was set to be thick enough (3.6 micrometers in this paper) with the doping concentration of about 2x1017 cm-3, and the optimal top cell thickness was suggested to be about 0.52 micrometers with doping concentration of about 1x1017 cm-3.

The absorption coefficient of the GaInP can be fitted by

$$
\alpha\_{\text{Gal}\,\text{nP}} = 5.5 \sqrt{(E - E\_{\text{g}})} + 1.5 \sqrt{(E - E\_{\text{g}} - 1)} \tag{19}
$$

The absorption coefficient of the GaInAs (with In content of about 0.01) can be fitted by

$$
\alpha\_{\text{Gal}\,nAs} = 3.3\sqrt{(E - E\_{\text{g}})} \tag{20}
$$

The direct gap absorption spectra of the bulk Ge was used for calculation

$$
\alpha\_{\rm Ge} = 1.9 \sqrt{(E - E\_g^{\rm I})} / E \tag{21}
$$

Where *E* is the photon energy and *Eg* the fundamental energy gap, both in eV, and *α* in 1/ micrometers.

The diffusion length, the diffusion coefficient and the nonradiative carrier lifetime are calcu‐ lated as a function of the doping concentration. The material parameters used for calculation are summarized in table 1.

#### **2.3. The effect of the interface recombination on the performance of GaInP/GaInAs/Ge tandem solar cell**

However, a high *Sp* also causes a reduction in the red response. In contrast, high *Sn* causes a reduction only in the red response (Figure 2 (a), Figure 3(a)), with almost no measurable ef‐

III-V Multi-Junction Solar Cells http://dx.doi.org/10.5772/50965 451

**Figure 4.** a) External quantum efficiency, and (b) integrated photocurrent density of the middle GaInAs cell for various

Once the photocurrents of the three sub-cells are calculated, the short circuit current of the tandem cell is set to be the smallest of these three photocurrents. The open-circuit voltage is set to be the voltage at which the magnitude of the dark currents equals the photocurrents. The corresponding I-V characteristics of the tandem cell are plotted in Figure 5. Among all the interfaces, recombination at the top cell emitter surface is most detrimental due to the considerable drop of the cell short circuit current and to a less extent to the associated reduc‐ tion in the cell voltage. While recombination effect at back interface of the bottom cell can be

**Figure 5.** I-V characteristics of the GaInP/ GaInAs/ Ge tandem cell under AM0 with a recombination velocity at the

fect in the blue response for a thick cell as shown in Figure 4 (a).

almost negligible because the base layer of the cell is thick enough.

interface recombination velocities.

indicated interface and zero elsewhere.

To have an analytical analysis, recombination velocity at only one interface among six inter‐ faces is assumed to have a non-zero value, which is 1х10<sup>6</sup> cm/s. Figure2~4 shows the total external quantum efficiencies *η* and the integrated photocurrent density *J ph* of the three sub-cells, calculated from Eqs. 2-5 with the constant parameters in table 1 and with varying values of *Sp* and *Sn* . The external quantum efficiency *η* , defined as the probability of collect‐ ing a photo carrier for each photon, is a function of wavelength, λ, because of the λ-depend‐ ence of the absorption coefficient, α. The photocurrent density *J ph* is obtained from the integral of the product of the *η* with the spectrum of interest. For large absorption coeffi‐ cients, a high *Sp* causes dramatic decrease in the blue response as shown in Figure 2 (a), Figure 3 (a) and Figure 4 (a).

**Figure 2.** a) External quantum efficiency, and (b) integrated photocurrent density of the top GaInP cell for various in‐ terface recombination velocities.

**Figure 3.** a) External quantum efficiency, and (b) integrated photocurrent density of the middle GaInAs cell for various interface recombination velocities.

However, a high *Sp* also causes a reduction in the red response. In contrast, high *Sn* causes a reduction only in the red response (Figure 2 (a), Figure 3(a)), with almost no measurable ef‐ fect in the blue response for a thick cell as shown in Figure 4 (a).

**2.3. The effect of the interface recombination on the performance of GaInP/GaInAs/Ge**

To have an analytical analysis, recombination velocity at only one interface among six inter‐ faces is assumed to have a non-zero value, which is 1х10<sup>6</sup> cm/s. Figure2~4 shows the total external quantum efficiencies *η* and the integrated photocurrent density *J ph* of the three sub-cells, calculated from Eqs. 2-5 with the constant parameters in table 1 and with varying values of *Sp* and *Sn* . The external quantum efficiency *η* , defined as the probability of collect‐ ing a photo carrier for each photon, is a function of wavelength, λ, because of the λ-depend‐ ence of the absorption coefficient, α. The photocurrent density *J ph* is obtained from the integral of the product of the *η* with the spectrum of interest. For large absorption coeffi‐ cients, a high *Sp* causes dramatic decrease in the blue response as shown in Figure 2 (a),

**Figure 2.** a) External quantum efficiency, and (b) integrated photocurrent density of the top GaInP cell for various in‐

**Figure 3.** a) External quantum efficiency, and (b) integrated photocurrent density of the middle GaInAs cell for various

**tandem solar cell**

450 Optoelectronics - Advanced Materials and Devices

Figure 3 (a) and Figure 4 (a).

terface recombination velocities.

interface recombination velocities.

**Figure 4.** a) External quantum efficiency, and (b) integrated photocurrent density of the middle GaInAs cell for various interface recombination velocities.

Once the photocurrents of the three sub-cells are calculated, the short circuit current of the tandem cell is set to be the smallest of these three photocurrents. The open-circuit voltage is set to be the voltage at which the magnitude of the dark currents equals the photocurrents. The corresponding I-V characteristics of the tandem cell are plotted in Figure 5. Among all the interfaces, recombination at the top cell emitter surface is most detrimental due to the considerable drop of the cell short circuit current and to a less extent to the associated reduc‐ tion in the cell voltage. While recombination effect at back interface of the bottom cell can be almost negligible because the base layer of the cell is thick enough.

**Figure 5.** I-V characteristics of the GaInP/ GaInAs/ Ge tandem cell under AM0 with a recombination velocity at the indicated interface and zero elsewhere.

### **2.4. Optimization of high efficiency GaInP/GaInAs/Ge multi-junction solar cells**

Lattice-matched GaInP/GaInAs/Ge triple-junction cells under investigation include a Ge bot‐ tom cell, a GaInAs middle cell and a GaInP top cell with energy gaps of 0.661, 1.405 and 1.85 eV, respectively. The Ge cell is built on the p-type initial substrate; the Ge base is 150 micro‐ meters thick with doping concentration of 6x1017 cm-3, and the Ge emitter is 300 nm thick with an n-type doping concentration of 1x1019 cm-3. The middle cell's base is set to be thick enough (3.6 micrometers in this paper) with doping concentration of 2x1017 cm-3, and its emitter is 100 nm thick with doping concentration of 1x1018 cm-3. The incident photon flux is taken from a newly proposed reference air mass zero (AM0) spectra (ASTM E-490). The antireflective coating used in simulation includes a 30 nm AlInP top window layer; ARC com‐ posed of 52 nm ZnS and 90 nm MgF2.

tandem cell for various top cell bases doping concentrations with thickness of 500 nm, when the top cell emitter thickness is set to 100 nm with doping concentration of 1x1018 cm-3. It is found that photocurrents strongly depend on top cell thickness, since the AM0 spectrum contains relatively more high-energy photons with energy greater than the GaInP top cell's energy gap, and photocurrents of triple-junction cells with a very thick top cell will general‐ ly be limited by the middle (GaInAs) cell. The tandem cell efficiency reaches the largest val‐ ue (31.27%) with the top cell base thickness of 500 nm, because the photocurrents of the top and middle cells almost match each other. Table 3 shows that higher doping concentration at the top cell base leads to a considerable increase of the cell voltage and a less drop of cell photocurrent. It can be deduced from table 3 that doping concentration at the top cell base should be optimized between 5 x1016 and 1x1017 cm-3 to obtain higher efficiency. In order to realize the values of the Figure-of-merits shown in table 2 and table 3, the external quantum efficiency of the top cell for various top cell base thicknesses and top cell base doping con‐ centrations are presented in Figure6 (a) and Figure6 (b), respectively. It is found that the ex‐ ternal quantum efficiency of the top cell increases with the increasing top cell base thickness (Figure6 (a)), while at short wavelengths, the efficiency increases with the increasing top cell

base doping concentration, at large wavelengths, decreases (Figure6 (b)).

**open-circuit voltage Voc (V)**

**Table 4.** Figure-of-merits of the tandem cell for various top cell emitter thickness.

concentration (b).

**emitter (nm)**

**top cell emitter thickness d-**

**Figure 6.** External quantum efficiency of the top cell for various top cell base thickness (a), and top cell base doping

d-emitter =50 nm 2.6680 0.01848 88.62% 31.98% d-emitter =100 nm 2.6667 0.01812 88.40% 31.27% d-emitter =150 nm 2.6710 0.01737 89.09% 30.25% d-emitter =200 nm 2.6707 0.01644 89.84% 28.87%

**short-circuit current**

**fill factor** **tandem cell efficiency**

III-V Multi-Junction Solar Cells http://dx.doi.org/10.5772/50965 453

**Jsc (A/cm2)**

It is at first assumed that recombination velocity for a top cell back surface is 1.3х10<sup>5</sup> cm/s, a middle cell back surface 105 cm/s and a top cell emitter surface 5.15х10<sup>4</sup> cm/s, while recom‐ bination velocities at the other three interfaces are zero. Then, the optimal top cell thickness and dopant profiles were obtained to meet high efficiency.


**Table 2.** Figure-of-merits of the tandem cell for various top cell base thickness.


**Table 3.** Figure-of-merits of the tandem cell for various top cell base doping concentration.

Table 2 presents the Figure-of-merits of the tandem cell for various top cell base thicknesses with doping concentration of 1x1017 cm-3, when the top cell emitter thickness is set to 100 nm with doping concentration of 1x1018 cm-3. Table 3 presents the Figure-of-merits of the tandem cell for various top cell bases doping concentrations with thickness of 500 nm, when the top cell emitter thickness is set to 100 nm with doping concentration of 1x1018 cm-3. It is found that photocurrents strongly depend on top cell thickness, since the AM0 spectrum contains relatively more high-energy photons with energy greater than the GaInP top cell's energy gap, and photocurrents of triple-junction cells with a very thick top cell will general‐ ly be limited by the middle (GaInAs) cell. The tandem cell efficiency reaches the largest val‐ ue (31.27%) with the top cell base thickness of 500 nm, because the photocurrents of the top and middle cells almost match each other. Table 3 shows that higher doping concentration at the top cell base leads to a considerable increase of the cell voltage and a less drop of cell photocurrent. It can be deduced from table 3 that doping concentration at the top cell base should be optimized between 5 x1016 and 1x1017 cm-3 to obtain higher efficiency. In order to realize the values of the Figure-of-merits shown in table 2 and table 3, the external quantum efficiency of the top cell for various top cell base thicknesses and top cell base doping con‐ centrations are presented in Figure6 (a) and Figure6 (b), respectively. It is found that the ex‐ ternal quantum efficiency of the top cell increases with the increasing top cell base thickness (Figure6 (a)), while at short wavelengths, the efficiency increases with the increasing top cell base doping concentration, at large wavelengths, decreases (Figure6 (b)).

**2.4. Optimization of high efficiency GaInP/GaInAs/Ge multi-junction solar cells**

It is at first assumed that recombination velocity for a top cell back surface is 1.3х10<sup>5</sup>

d-base=400 nm 2.6660 0.01720 90.56% 30.40% d-base=450 nm 2.6664 0.01768 89.82% 30.99% d-base=500 nm 2.6667 0.01812 88.40% 31.27% d-base=55 0nm 2.6669 0.01777 89.82% 31.05%

NA-base =1х1016 2.6095 0.01816 88.62% 30.74% NA-base =5х1016 2.6503 0.01814 89.15% 31.32% NA-base =1х1017 2.6667 0.01812 88.40% 31.27% NA-base =5х1017 2.6981 0.01796 88.24% 31.19%

middle cell back surface 105 cm/s and a top cell emitter surface 5.15х10<sup>4</sup> cm/s, while recom‐ bination velocities at the other three interfaces are zero. Then, the optimal top cell thickness

> **short-circuit current Jsc (A/cm2)**

**short-circuit current Jsc (A/**

**cm2)**

Table 2 presents the Figure-of-merits of the tandem cell for various top cell base thicknesses with doping concentration of 1x1017 cm-3, when the top cell emitter thickness is set to 100 nm with doping concentration of 1x1018 cm-3. Table 3 presents the Figure-of-merits of the

**fill factor**

**fill factor** cm/s, a

**tandem cell efficiency**

**tandem cell efficiency**

posed of 52 nm ZnS and 90 nm MgF2.

452 Optoelectronics - Advanced Materials and Devices

**top cell base thickness d-base**

**top cell base doping concentration (1/cm3)**

**(nm)**

and dopant profiles were obtained to meet high efficiency.

**open-circuit voltage Voc (V)**

**Table 2.** Figure-of-merits of the tandem cell for various top cell base thickness.

**open-circuit voltage Voc (V)**

**Table 3.** Figure-of-merits of the tandem cell for various top cell base doping concentration.

Lattice-matched GaInP/GaInAs/Ge triple-junction cells under investigation include a Ge bot‐ tom cell, a GaInAs middle cell and a GaInP top cell with energy gaps of 0.661, 1.405 and 1.85 eV, respectively. The Ge cell is built on the p-type initial substrate; the Ge base is 150 micro‐ meters thick with doping concentration of 6x1017 cm-3, and the Ge emitter is 300 nm thick with an n-type doping concentration of 1x1019 cm-3. The middle cell's base is set to be thick enough (3.6 micrometers in this paper) with doping concentration of 2x1017 cm-3, and its emitter is 100 nm thick with doping concentration of 1x1018 cm-3. The incident photon flux is taken from a newly proposed reference air mass zero (AM0) spectra (ASTM E-490). The antireflective coating used in simulation includes a 30 nm AlInP top window layer; ARC com‐

**Figure 6.** External quantum efficiency of the top cell for various top cell base thickness (a), and top cell base doping concentration (b).


**Table 4.** Figure-of-merits of the tandem cell for various top cell emitter thickness.

The Ge sub-cell is an important part of the structure of this cell, contributing 10% or more of the total cell efficiency [13]. The Ge junction is formed during III - V /Ge interface epitaxy. Group V elements such as P and As are n-type dopants in Ge, so the emitter of Ge junction was formed by diffusion of V elements during the deposition of III - V epilayers. In addition, the structure of Ge is different from the III - V materials such as GaAs and GaInP, the con‐ nection between Ge substrate and buffer layer or initial layer is important to the growth quality on buffer layer and the performance of Ge sub-cell. In this chapter, based on plenty of experiments, GaInP is selected as a suitable buffer material to be grown between the sub‐ strate and the active region of the device. Several researches on III - V materials grown on pdoped Ge substrate have indicated that the bottom Ge cell efficiency decreases as the thickness of the emitter increases, mainly owing to the lowering of the short circuit current. For this reason, GaInP is an optimized option with smaller diffusion length than GaAs. In

addition, GaInP is also an appropriate material for the window layer of Ge junction.

the energy gap in the middle cell.

high doping AlGaAs/GaAs tunnel junctions.

cation operate at high current densities higher than 10A/cm2

The electrochemical capacitance-voltage results of GaInP initial layer grown on Ge indicate that the diffusion length of P is about 200 nm, when a thin Ge emitter for excellent perform‐ ance of Ge sub-cell is fabricated. In the past, GaAs was employed as the middle cell material, and the 0.08% lattice-mismatch between GaAs and Ge was thought to be negligibly. To ob‐ tain enough current matched to the top cell, the middle cell was often designed to be 3~4 micrometers thick, but misfit-dislocations were generated in thick GaAs layers and deterio‐ rated cell performance [5]. By adding about 1% indium into the GaAs cell layers, all cell lay‐ ers are lattice-matched precisely to the Ge substrate. Application of InGaAs middle cell to lattice-match Ge substrates has demonstrated to be able to increase open-circuit voltage (Voc) due to lattice-matching and short-circuit current density (Jsc) due to the decrease of

The Ga0.49In0.51P/Ga0.99In0.01As/Ge multi-junction solar cells for terrestrial concentrator appli‐

to the tunnel diode structures that are used for the series connection of the sub-cells. So the tunnel junction (TJ) growth is one of the most important issues affecting multi-junction solar cell performance. The problems of TJ growth are related to obtaining transparent and uni‐ formly highly doped layer without any degradation of surface morphology [14]. The thick‐ ness of each side of the TJ junction has to be in the order of tens of nanometres, while the required doping has to be around 1019~1020 cm-3. The reaching of the high doping level re‐ quires very different growth temperatures, in order to obtain an abrupt doping profile. In this experiment, the growth of tunnel junction was carried out at temperature of 600 °C which is about 50 °C lower than the growth temperature of other layers. DETe and CCl4 were used as N-type dopant and P-type dopant respectively to fabricate small thickness,

GaInP lattice-matched to GaAs exhibits anomalous changes in the energy gap, depending on the growth conditions and the substrate misorientation [15]. These changes are the results of the spontaneous ordering during the growth of the cation-site elements (Ga and In) in planes parallel to the (111). One of changes is a lowering of the energy gap of the material, whose exact value depends on the degree of ordering. It appears to be the 100 meV reductions. The

. This brings specific challenges

III-V Multi-Junction Solar Cells http://dx.doi.org/10.5772/50965 455

**Figure 7.** The external quantum efficiency of the top cell for various top cell emitter thickness (a), and top cell emitter doping concentration (b).


**Table 5.** Figure-of-merits of the tandem cell for various top cell emitters doping concentration.
