**2. Open circuit voltage and carrier inversion**

The output performance of the solar cells can be described by *V*OC, *J*SC and FF. All such parameters are linked with the *η* and define the overall output performance of solar cells. While the main aim of this chapter is to describe the role of heterointerface and inversion at the a-Si:H/c-Si, we will focus on *V*OC which is strongly affected by recombination properties and carrier transport in the solar cell. The *V*OC for SHJp solar cells can be expressed by the analytical model as [28, 29]

$$V\_{\rm OC} = \frac{E\_{\rm g-Si} - \mathcal{S}\_{\rm Si(\rho)}}{q} - \frac{kT}{q} \ln\left(\frac{\frac{D\_{\rm p}}{D\_{\rm p}} + S\_{\rm p}}{\Delta n}\right) \tag{1}$$

Similarly for SHJn, the *V*OC is expressed as

$$V\_{\rm OC} = \frac{E\_{g-Sl} - \mathcal{S}\_{\rm Sl}(n)}{q} - \frac{kT}{q} \ln\left(\frac{N\_V}{\Delta p} \cdot \frac{\frac{D\_\mathbf{n}}{L\_\mathbf{n}} + S\_\mathbf{n}}{\frac{D\_\mathbf{n}}{L\_\mathbf{n}}}\right) \tag{2}$$

Symbols in the above equations denote: *T* is the temperature, *q* is the elementary charge, *k* is the Boltzmann constant, *E*g-Si is the band gap of c-Si, *N*C and *N*V are the effective densities of states in the conduction band of c-Si, *δSi*(*p*) and *δSi*(*n*) are the dopant activation energies of c-Si substrate with p-type and n-type doping, respectively, *L*p and *L*<sup>n</sup> are diffusion lengths for holes and electrons, respectively, and *D*p and *D*n are diffusion constants for holes and electrons. Further symbols denote the interface recombination velocities for holes p <sup>=</sup> p it and electrons n <sup>=</sup> nit, where *C*p and *C*n are the capture rate coefficients for holes and electrons, respectively, and *D*it is the interface defect density. The Δ = Δ <sup>=</sup> . eff in the equations denotes the excess carrier concentration, where *g* is the average photo-generation of the electron-hole pairs in c-Si and *τ*eff is the effective lifetime of the excess carries.

From the above equations it is apparent that *V*OC depends on Δ*n*, which is determined by *τ*eff and *g. g* is related with the illumination intensity. *τ*eff is determined by the recombination velocities *S*p and *S*n and thus by the defect state density at the a-Si:H/c-Si heterointerface, *D*it recombination at the rear surface and recombination in the c-Si substrate. The recombination in the c-Si substrate is not the subject of this chapter, instead of this, we focus our attention to the a-Si:H/c-Si interface and inversion layer formed at c-Si surface of this interface. In the case of low recombination in the bulk and at the back surface of c-Si, the main recombination path is at the heterointerface. For such a case, the saturation current of the SHJ, *J*sat, is determined by the saturation current of interface recombination *J*sat-it, which is for SHJp determined by the interface recombination velocity *S*n and holes concentration at the heterointerface *p*it as

$$j\_{\rm sat-it} = qS\_{\rm p}p\_{\rm it} \tag{3}$$

Similarly, by considering interface concentration of electrons *n*it for SHJn it can be written

$$j\_{\rm sat-it} = qS\_{\rm n}n\_{\rm it} \tag{4}$$

Silicon Heterojunction Solar Cells: The Key Role of Heterointerfaces and their Impact on the Performance http://dx.doi.org/10.5772/65020 75

By taking into account the equation for *V*OC

$$V\_{\rm OC} = \frac{AkT}{q} \ln\left(\frac{j\_{\rm SC}}{j\_{\rm sat}}\right) \tag{5}$$

and substituting *J*sat-it as a saturation current, it is possible to write equation which determines the *V*OC as a function of the interface recombination velocity and effective barrier for recombination at the heterointerface *Φ*B [30]

$$V\_{\rm OC} = \frac{\Phi\_{\rm B}}{q} - \frac{AkT}{q} \ln\left(\frac{qN\_{\rm V}S\_{\rm p}}{j\_{\rm SC}}\right) \tag{6}$$

for SHJp and

( )

( )

d-

<sup>n</sup> OC <sup>n</sup>

ln . *g Si Si n <sup>V</sup> <sup>D</sup> <sup>S</sup> <sup>E</sup> kT <sup>N</sup> <sup>L</sup> <sup>V</sup>*

*q qp D*

Symbols in the above equations denote: *T* is the temperature, *q* is the elementary charge, *k* is the Boltzmann constant, *E*g-Si is the band gap of c-Si, *N*C and *N*V are the effective densities of

and *δSi*(*n*)

substrate with p-type and n-type doping, respectively, *L*p and *L*<sup>n</sup> are diffusion lengths for holes and electrons, respectively, and *D*p and *D*n are diffusion constants for holes and electrons.

electrons n <sup>=</sup> nit, where *C*p and *C*n are the capture rate coefficients for holes and electrons, respectively, and *D*it is the interface defect density. The Δ = Δ <sup>=</sup> . eff in the equations denotes the excess carrier concentration, where *g* is the average photo-generation of the

From the above equations it is apparent that *V*OC depends on Δ*n*, which is determined by *τ*eff and *g. g* is related with the illumination intensity. *τ*eff is determined by the recombination velocities *S*p and *S*n and thus by the defect state density at the a-Si:H/c-Si heterointerface, *D*it recombination at the rear surface and recombination in the c-Si substrate. The recombination in the c-Si substrate is not the subject of this chapter, instead of this, we focus our attention to the a-Si:H/c-Si interface and inversion layer formed at c-Si surface of this interface. In the case of low recombination in the bulk and at the back surface of c-Si, the main recombination path is at the heterointerface. For such a case, the saturation current of the SHJ, *J*sat, is determined by the saturation current of interface recombination *J*sat-it, which is for SHJp determined by the interface recombination velocity *S*n and holes concentration at the heterointerface *p*it as

sat it p it

Similarly, by considering interface concentration of electrons *n*it for SHJn it can be written

sat it n it

Further symbols denote the interface recombination velocities for holes p <sup>=</sup> p

electron-hole pairs in c-Si and *τ*eff is the effective lifetime of the excess carries.

= - ç ÷

æ ö <sup>+</sup> - ç ÷

d-

ln . *g Si Si p <sup>C</sup>*

*<sup>S</sup> <sup>E</sup> kT <sup>N</sup> <sup>L</sup>*

ç ÷ <sup>+</sup> - ç ÷ <sup>=</sup> - ç ÷ <sup>D</sup> ç ÷

*q qn D*

OC

*V*

74 Nanostructured Solar Cells

Similarly for SHJn, the *V*OC is expressed as

states in the conduction band of c-Si, *δSi*(*p*)

p

*D*

æ ö

p

è ø

p p

<sup>n</sup> <sup>n</sup>

n

are the dopant activation energies of c-Si

*j qS p* - = (3)

*j qS n* - = (4)

*L*

ç ÷ D ç ÷ è ø

*L*

p

(1)

(2)

it and

$$V\_{\rm OC} = \frac{\Phi\_{\rm B}}{q} - \frac{AkT}{q} \ln\left(\frac{qN\_{\rm C}S\_{\rm n}}{j\_{\rm SC}}\right) \tag{7}$$

for SHJn, where *A* represents the diode ideality factor. From the above equation it is obvious that *V*OC is determined by the *Φ*B which should have a high value to obtain high *V*OC. **Figure 2** show the band diagram of SHJn and SHJp with *Φ*B at the heterointerfaces, respectively. In SHJ solar cells the c-Si surface at the heterointerface is inverted or strongly inverted, forming an inversion layer with a high concentration of minority carriers [31, 32] at the heterointerface. From the band diagram, the rate of the inversion is determined by the bending of bands at the surface of c-Si and can be expressed as a distance of the Fermi level from the conduction band level at the heterointerface, i = C – f . It is obvious that the car-

**Figure 2.** Band diagram of (left) SHJn and (right) SHJp structures with sketched barrier for interface recombination *Φ*<sup>B</sup> at interface defects *D*it.

rier inversion is linked with *Φ*B thus directly related to *V*OC. For both SHJ, high carrier inversion is required to obtain high value of *Φ*<sup>B</sup> and thus high *V*OC. Eqs. (1) and (2) do not take into account the influence of a-Si:H layer and indicate that *V*OC depends on the illumination intensity, recombination properties at the heterointerface, recombination at the rear surface and in the c-Si substrate, and on the dopant activation energy of the c-Si substrate. The properties of the emitter seem to play no role in the *V*OC. In fact, the parameters like doping, defect density and affinity (or band offset with c-Si) have no direct influence on the *V*OC. However, all of these parameters affect the charge properties of the space charge region (SCR) of SHJ junction and thus carrier inversion at the heterointerface and consequently *V*OC. In the following sections, we will describe by means of simulation various SHJ solar cell properties which affect carrier inversion and *V*OC.
