**2. Background theory: The p-n junction**

Photonic device (solar cells included) operation is based on a p-n junction: two regions of the semiconducting material doped p and n type respectively and brought together in contact form a p-n junction. At thermal equilibrium, the p-n dope bulk semiconducting crystal, in order to keep its equilibrium, develops an internal field and develops its own built-in potential; the latter is total due to p- and n-type carrier migration across the junction.

Donor and acceptor atoms embedded in the lattice of the host material provide electrons and holes (as potential current carriers) that are free to wander in the crystal. In principle these carriers move randomly in the lattice, however, guiding these carriers accordingly could lead to non-zero currents coming off such semiconductors, and therefore to current producing devices. A semiconductor sample doped with donors and acceptors becomes a pn junction and therefore a device with two regions tending to overlap at their boundary.

High Efficiency Solar Cells via Tuned Superlattice Structures: Beyond 42.2% 327

Fig. 2. Potential V as a function of x across the depletion region. Note (a) the two branches of V across both sides of the junction's boundary (x = 0) in accordance with (4) and (5) (b) the built-in voltage Vbi at the right edge of the junction [1, 2, 3]. Note also that built-in voltage is

It is a straightforward matter to produce explicit results about widths in the junction area (w, xn, -xp) in terms of device doping levels and built-in voltage values. The built-in voltage

> 2 2 ()( ) <sup>2</sup> *bi <sup>n</sup> an d p <sup>q</sup> V V x x Nx Nx*

The fundamental current equation for p-n junctions is derived based on considering that the built-in voltage is reduced down to Vbi – Va , by the forward bias voltage Va, helping majority carriers to escape and diffuse in the neighboring regions while, once electrons and holes reach the edges of the depletion region to the p and n regions, they diffuse accordingly according to a decaying exponential law of the type exp(x/Ln,p); the latter includes distance x and the diffusion length for electrons and/or holes respectively. Excess minority carriers

> / ( )/ ( ) ( 1) *x L xx L no*

> > / ( ) ( 1) *x L po*

*p*

*px p e e*

*p np*

*diffuse* in both regions according to the following expressions:

(6)

(7)

*nx n e* (8)

normally computed as shown in the inset.

is determined from (4) at x = xn:

**3. Diode currents** 

If the interface is at (say) x = 0 position, free electrons and free holes diffuse through the interface and inevitably form space charge regions as shown figure 1 below:

Fig. 1. pn-junction (e.g. of a Si sample) with the depletion W region shown: both sides of the interface are shown, with their space charge distributions respectively.

A static electric field develops at the interface (figure above) emanating from the (+) region and prohibiting respective carriers to further access the PN regions. From basic pnjunction theory, we can solve for the electric field and the potential developed by means of Poisson's equation. If the limits of the depletion region are –xp and xn (W = xp + xn) respectively, we can derive expressions for both field and potential developed at the junction:

$$E(\mathbf{x}) = \frac{qN\_d}{\mathcal{E}}(\mathbf{x} - \mathbf{x}\_s) \qquad \qquad 0 < \mathbf{x} < \mathbf{x}\_s \tag{2}$$

$$E(\mathbf{x}) = -\frac{qN\_s}{\varepsilon}(\mathbf{x} + \mathbf{x}\_\rho) \qquad \quad -\mathbf{x} \quad < \mathbf{x} < \mathbf{0} \tag{3}$$

Where maximum field value is *E*max = *E*(at x = 0) = - *(q Nd /) xn*; q is the electronic charge, Nd,a stands for donor and acceptor atom concentrations (per volume) respectively, is the total sample's dielectric constant (or the product of the relative times the free space dielectric constants, e.g. r = 11.7 for Si). Based on expressions (2, 3) and on the fact that potential generated at the junction is the negative integral of the electric field across the depletion region, we can in principle derive the potential V(x) across the junction: it can be shown that V(x) is as follows:

$$V(\mathbf{x}) = \frac{qN\_s}{2\varepsilon}(\mathbf{x} + \mathbf{x}\_\rho)^\natural; \text{In the p-region, and}\tag{4}$$

$$V(\mathbf{x}) = -\frac{qN\_d}{2\varepsilon}[(\mathbf{x} - \mathbf{x}\_\*)^2 - \mathbf{x}\_\*^2] + \frac{qN\_o}{2\varepsilon}\mathbf{x}\_\rho^2; \text{ In the n-region and}\tag{5}$$

Fig. 2. Potential V as a function of x across the depletion region. Note (a) the two branches of V across both sides of the junction's boundary (x = 0) in accordance with (4) and (5) (b) the built-in voltage Vbi at the right edge of the junction [1, 2, 3]. Note also that built-in voltage is normally computed as shown in the inset.

It is a straightforward matter to produce explicit results about widths in the junction area (w, xn, -xp) in terms of device doping levels and built-in voltage values. The built-in voltage is determined from (4) at x = xn:

$$V\_{\boldsymbol{\kappa}} = V(\boldsymbol{\kappa} = \boldsymbol{\kappa}\_{\boldsymbol{\kappa}}) = \frac{q}{2\mathcal{E}} (N\_{\boldsymbol{\kappa}}\boldsymbol{\kappa}\_{\boldsymbol{\kappa}}^2 + N\_{\boldsymbol{\kappa}}\boldsymbol{\kappa}\_{\boldsymbol{\rho}}^2) \tag{6}$$

### **3. Diode currents**

326 Solar Cells – New Aspects and Solutions

If the interface is at (say) x = 0 position, free electrons and free holes diffuse through the

E-field

Fig. 1. pn-junction (e.g. of a Si sample) with the depletion W region shown: both sides of the

W

P-type (-) (+) N-type

A static electric field develops at the interface (figure above) emanating from the (+) region and prohibiting respective carriers to further access the PN regions. From basic pnjunction theory, we can solve for the electric field and the potential developed by means of Poisson's equation. If the limits of the depletion region are –xp and xn (W = xp + xn) respectively, we can derive expressions for both field and potential developed at the

*n*

*p*

Nd,a stands for donor and acceptor atom concentrations (per volume) respectively, is the total sample's dielectric constant (or the product of the relative times the free space dielectric constants, e.g. r = 11.7 for Si). Based on expressions (2, 3) and on the fact that potential generated at the junction is the negative integral of the electric field across the depletion region, we can in principle derive the potential V(x) across the junction: it can be shown that

<sup>0</sup> *<sup>n</sup> <sup>x</sup> <sup>x</sup>* (2)

*x x* 0 (3)

*) xn*; q is the electronic charge,

; In the p-region, and (4)

; In the n-region and (5)

interface are shown, with their space charge distributions respectively.

() ( ) *<sup>d</sup>*

() ( ) *<sup>a</sup>*

*qN Ex x x* 

<sup>2</sup> () ( ) <sup>2</sup> *a*

22 2 ( ) [( ) ] 2 2 *d a*

*qN qN Vx x x x x*

*qN Vx x x* 

*p*

*nn p*

 

Where maximum field value is *E*max = *E*(at x = 0) = - *(q Nd /*

*qN Ex x x* 

junction:

V(x) is as follows:

interface and inevitably form space charge regions as shown figure 1 below:

The fundamental current equation for p-n junctions is derived based on considering that the built-in voltage is reduced down to Vbi – Va , by the forward bias voltage Va, helping majority carriers to escape and diffuse in the neighboring regions while, once electrons and holes reach the edges of the depletion region to the p and n regions, they diffuse accordingly according to a decaying exponential law of the type exp(x/Ln,p); the latter includes distance x and the diffusion length for electrons and/or holes respectively. Excess minority carriers *diffuse* in both regions according to the following expressions:

$$\delta \, p(\mathbf{x}) = p\_{\,\,no}(e^{\mathbf{x}^{\prime \ell}p} - \mathbf{l})e^{(\mathbf{x} + \mathbf{z}\_{\mathbb{R}}) \cdot \mathbf{l}} \, \tag{7}$$

$$
\delta \mathcal{S} \mathfrak{n}(\mathbf{x}) = \mathfrak{n}\_{\mu \flat} (e^{\pi^{\*L} p} - \mathbf{l}) \tag{8}
$$

High Efficiency Solar Cells via Tuned Superlattice Structures: Beyond 42.2% 329

Fig. 3. Typical modeling geometry of a solar cell: w is the depletion width, J is the exact interface, L is the width of the p-region and d is the n-region (window layer). Note that the

W

N-region P-region

(1 )( / ) ( ) exp( / ) 1( )

*px C xL <sup>e</sup> <sup>L</sup>* 

2

2

*p RL D*

*p p*

*n*

Solution of (18) is of similar kind with (12) along with boundary conditions (17):

*L L*

The surface recombination velocity sn at the edge of the p-region is

2

*dn n n*

*dx L*

*ph p p x d*

2

2

*p RL D*

( )

( ) tanh( ) cosh( / ) cosh( / )

*p p p p*

*<sup>d</sup> L sL p d <sup>e</sup> L dL D dL*

2 2 1 (1 )( / ) ( )1

() ( ) *n x L W n p po dn D sn n*

<sup>2</sup> (1 ) 0 *p p po x d*

(1 )( / ) ( ) cosh( / ) sinh( / ) 1( )

*nx A x L B x L <sup>e</sup> <sup>L</sup>*

*n n*

*ph p p*

*F RL D*

*ph p p x d*

(15)

*p p p d*

*dx* (17)

(18)

2

2

*n RL D*

*ph n n x d*

( )

( )

(19)

*ph*

*R e*

 (16)

( )

x = w x = w+L

Minority holes generated in the window layer (x from –d to 0) are:

*n p*

(1 )( / ) (0) 1( )

*p e <sup>L</sup>*

Maximum hole- current density generated in the n-region is:

n-region is the window for solar photons.

x= - d x= 0

*n*

( 0)

*p p*

The diffusion equation reads as follows:

*J x qD*

Note that at x = 0:

Where pno is holes in the n-region, Lp is the diffusion length of holes in the n-region, and where p represents excess holes in the n-region. Diffusion currents can be calculated by means of the diffusion equation along with suitable boundary conditions:

$$J\_{\rho} = -qD\_{\rho} \frac{d\delta \, p(\mathbf{x})}{d\mathbf{x}} \bigg| \quad \text{ ( $\mathbf{x} = \mathbf{x}\_{n}$ )}\tag{9}$$

$$Jn = qD\_\ast \frac{d\delta \, p(\mathbf{x})}{d\mathbf{x}}\Big|\_{} \quad \text{( $\mathbf{x} = -\mathbf{x}\_\mathbf{p}$ )}\tag{10}$$

Based on the above expressions, current density of the p-n junction due to a forward bias Va is found to be as follows (see also (1)):

$$J = J\_\circ(e^{\mathbb{I}[a \cap \mathbb{I}]} - \mathbb{I})\tag{11}$$

(Where Vt is the thermal voltage (kT/q))

### **4. p-n junctions as solar cells**

Fundamentally, solar cell modeling correlates incident solar photon fluxph (# of photons cm-2 s1) with generation and recombination carrier rates in the interior of the device. Photogenerated concentrations of diffusing carriers are typically modeled through the diffusion equation (under appropriate boundary conditions):

$$\frac{d^z \delta \, p\_s}{d\alpha^2} - \frac{\delta \, p\_s}{L\_\rho} + \alpha(1 - R)\Phi\_{ph}e^{-a(s+d)} = 0\tag{12}$$

Photon-collection efficiency is usually defined as the ratio of total current over solar photoflux (cm-2 s-1):

$$\eta\_{col} = \frac{J\_{\phantom{p}} + J\_{\phantom{s}} - J\_{\phantom{m}}}{q\Phi\_{\phantom{p}\flat}} \tag{13}$$

The numerator in (13) is total photo-induced current in the p and n-regions minus recombination current. Boundary conditions include continuity of carrier concentrations at the junction x (j), and the dependence of the first derivative of carrier concentration on recombination velocity sp, at the edge of the window layer as shown in the figure below:

$$(\frac{d\delta\,p}{d\mathbf{x}})\_{x=d} = \frac{s\_p}{D\_p}(p(-d) - p\_{\text{no}}) \equiv \frac{s\_p}{D\_p}(p(-d))\tag{14}$$

Figure-3 shows a generally accepted modeling geometry of a p-n junction solar cell. These two regions are separated by the depletion region (of thickness w): majority electrons from the n-region migrate to the p-region, and majority holes reciprocate from the latter region.

Where pno is holes in the n-region, Lp is the diffusion length of holes in the n-region, and where p represents excess holes in the n-region. Diffusion currents can be calculated by

( )

( )

Based on the above expressions, current density of the p-n junction due to a forward bias Va

Fundamentally, solar cell modeling correlates incident solar photon fluxph (# of photons cm-2 s1) with generation and recombination carrier rates in the interior of the device. Photogenerated concentrations of diffusing carriers are typically modeled through the diffusion

<sup>2</sup> (1 ) 0 *n n x d*

*p n rec*

*J J J q*

The numerator in (13) is total photo-induced current in the p and n-regions minus recombination current. Boundary conditions include continuity of carrier concentrations at the junction x (j), and the dependence of the first derivative of carrier concentration on recombination velocity sp, at the edge of the window layer as shown in

> ( ) ( ( ) ) ( ( )) *p p x d no*

Figure-3 shows a generally accepted modeling geometry of a p-n junction solar cell. These two regions are separated by the depletion region (of thickness w): majority electrons from the n-region migrate to the p-region, and majority holes reciprocate from the latter

*p p d p s s pd p pd dx D <sup>D</sup>*

*ph*

Photon-collection efficiency is usually defined as the ratio of total current over solar photo-

*p dp p R e*

*col*

*o*

/ ) ( 1) *V V*

(x = xn) (9)

(x = -xp) (10)

*a t J Je* (11)

( )

(12)

(13)

(14)

*ph*

means of the diffusion equation along with suitable boundary conditions:

*p p d px J qD dx* 

is found to be as follows (see also (1)):

(Where Vt is the thermal voltage (kT/q))

equation (under appropriate boundary conditions):

2

 *dx L* 

 

**4. p-n junctions as solar cells** 

flux (cm-2 s-1):

the figure below:

region.

*n d px Jn qD dx* 

Fig. 3. Typical modeling geometry of a solar cell: w is the depletion width, J is the exact interface, L is the width of the p-region and d is the n-region (window layer). Note that the n-region is the window for solar photons.

Minority holes generated in the window layer (x from –d to 0) are:

$$\delta \mathcal{P}\_{\mathbf{r}}(\mathbf{x}) = C \exp(-\mathbf{x} \mid L\_{\rho}) + \frac{\alpha \Phi\_{\rho k} (\mathbf{l} - R) (L\_{\rho}^2 \mid D\_{\rho})}{\mathbf{l} - (\alpha L\_{\rho})^2} e^{-a(\mathbf{x} + \mathbf{d})} \tag{15}$$

Note that at x = 0: 2 ( ) 2 (1 )( / ) (0) 1( ) *ph p p x d n p RL D p e <sup>L</sup>* 

Maximum hole- current density generated in the n-region is:

$$J\_{\boldsymbol{\rho}}(\mathbf{x}=\mathbf{0})=-qD\_{\boldsymbol{\rho}}\times\begin{bmatrix}\frac{1}{L\_{\boldsymbol{\rho}}}\left(\frac{aF\_{\rho\mathbf{i}}(\mathbf{I}-\mathbf{R})(L\_{\boldsymbol{\rho}}^{\perp}/D\_{\boldsymbol{\rho}})}{(aL\_{\boldsymbol{\rho}})^{2}-1}\right)\times\\\left[\tanh(\frac{d}{L\_{\boldsymbol{\rho}}})+\frac{aL\_{\boldsymbol{\rho}}}{\cosh(d/L\_{\boldsymbol{\rho}})}+\frac{s\_{\boldsymbol{\rho}}L\_{\boldsymbol{\rho}}}{D\_{\boldsymbol{\rho}}}\times\frac{p(-d)}{\cosh(d/L\_{\boldsymbol{\rho}})}\right]+ae^{-ad}\end{bmatrix}\tag{16}$$

The surface recombination velocity sn at the edge of the p-region is

$$-D\_{\pi}(\frac{dn}{d\mathfrak{x}})\_{x=L+W} = \mathfrak{s}\_{\pi}(n\_{\rho} - n\_{\rho\circ})\tag{17}$$

The diffusion equation reads as follows:

$$\frac{d^2 n\_{\rho}}{d\mathbf{x}^2} - \frac{n\_{\rho} - n\_{\rho o}}{L\_{\star}} + \alpha (\mathbf{l} - R) \Phi\_{\rho b} e^{-\alpha (z+d)} = \mathbf{0} \tag{18}$$

Solution of (18) is of similar kind with (12) along with boundary conditions (17):

$$\delta n(\mathbf{x}) = A \cosh(\mathbf{x} \mid L\_{\mathbf{s}}) + B \sinh(\mathbf{x} \mid L\_{\mathbf{s}}) + \frac{a \Phi\_{ph}(\mathbf{l} - R)(L\_{\mathbf{s}}^2 \mid D\_{\mathbf{s}})}{1 - (aL\_{\mathbf{s}})^2} e^{\mathbf{e}(\mathbf{z} + \mathbf{d})} \tag{19}$$

High Efficiency Solar Cells via Tuned Superlattice Structures: Beyond 42.2% 331

photons with energies higher than the band gap of the host material. The figure below depicts an intrinsic multi-quantum well area, where discrete energy levels cause a widening

Fig. 4. Detail from a superlattice structure (typically GaAs/alloy and GaAs/Ge (as proposed in this study). The dashed line represents the Fermi level at thermal equilibrium. The optical

Fig. 5. A p-i-n GaAs/alloy superlattice developed in the mid-region of a pin cell: the middle section depicts: the top layer (blue) is the wide gap alloy (e.g. AlAs) and the bottom layer is GaAs (host material); GaAs is also grown in the superlattice as the low gap medium.

Such a cell design (shown above) is an expanded p-n junction with a wide superlattice midregion occupying the intrinsic or low doped region between p and n. To reduce cost such a structure can be compromised by inserting a short period tuned superlattice as a small

Modeling of the top region may be performed in two ways, by considering the equivalent circuit of the device and/or by solving for excess carriers and subsequent electric currents and current densities in the solid state. In this brief outline we are considering the first

of the host material's gap (commonly GaAs with gap at 1.42 eV).

gap can be tuned to desired energy values.

percentage of the device as a total.

**6. Top cell (AlAs/GaAs)** 

Figure 5 shows a superlattice covering the mid region of a pin cell:

The total current out of the cell is the sum of all currents minus recombination components from each region, especially recombination at the w region. Excess carriers in solar cells (as in any photonic device) are minority electrons and holes in the p and n regions respectively. When a cell is illuminated, solar photons excite electron hole pairs in all regions: the p-, nand depletion regions. The latter may be become of great significance for the following reason: excited electrons and holes do split away from each other due to the existing electrostatic field. This means that these excess carriers will reach the edges of the depletion region in a very short time. Note also that typically, mean diffusion lengths of these carriers are much longer than t he actual width of the depletion area (even in pin devices). This makes the depletion region especially attractive for illumination: electrons and holes will separate from each other quickly, and they will diffuse in the bulk parts of the cell very fast assisted by the electrostatic field. In addition, space availability in the mid-region provides a chance for excess layer s that can be tuned to desired solar photons for subsequent absorption, thus enhancing device performance. This is why multi-layers are used in the intrinsic region (long depletion region in p-n junctions). If tuned quantum wells are grown somewhere in the middle, incident solar illumination will push electrons in the quantum wells and to tunneling or thermionic escape. The notion of additional band gaps integrated in the intrinsic region has been adopted successfully recently. For instance, successful cells with more than one band gaps have been designed and realized, where two or three cells are connected in series forming tandem cells with the advantage of voltage increase. This is possible due to the series connection of the tandem cells. Tandems provide excess voltage but they lack in current, in other words, due to the differences of the layers involved, current matching will be enforced due to the series connection. If these structures can ensure relatively high current outputs, then, along with increased voltage one should expect efficiency improvements. In the next we outline the behavior of a cell in tandem: top cell of AlAs/GaAs and bottom cell of a pin GaAs/Ge/Alloy for long wavelengths.
