High efficiency thin-film crystalline Si/Ge tandem solar cell

G. Sun; F. Chang; R. A. Soref

doi:10.1364/OE.18.003746

1. Introduction

The relatively weak power of sunlight limits the power output from a solar cell, which in turn increases its cost per unit power delivered. Solar-concentration systems have the potential to reduce the cost of solar-to-electricity conversion by using inexpensive lenses or mirrors to direct highly concentrated sunlight onto a small-area solar cell. But the “merit” of a photovoltaic (PV) cell depends upon the PV material manufacturing costs as well as the cell’s efficiency. Solar cells made of III-V semiconductor compounds have higher efficiency but are typically more expensive than those made of silicon, motivating a group IV solution. In this paper, we propose a solar concentrator that uses a tandem solar cell made of group-IV PV material in which a Si junction is stacked on top of a Ge junction. Such a solar structure is capable of delivering significantly higher efficiency than the single-junction all-Si solar cell.

Crystalline Ge is in many ways a better solar photovoltaic (PV) material than crystalline Si because Ge’s optical absorption has a wider spectral overlap with the solar irradiance spectrum (Ge covers the 300 to 1600 nm wavelength range compared with the 300 to 1060 nm coverage of Si), Ge has a steeper absorption edge than Si since its bandgap is almost direct at 0.66 eV, and the optical absorption coefficients of Ge are generally higher than those of Si in the range of interest. In addition, Ge is known to be an excellent bottom junction material in a multi-junction solar cell [1]. For these reasons, we are proposing a tandem PV cell comprised of a c-Si pn junction on top of a c-Ge pn junction with a tunnel hetero-diode in between. Matching the photocurrent in Ge to the photocurrent in Si quickly points to a cell structure consisting mostly of Ge with very thin Si. Our simulations predict conversion efficiencies that are significantly higher than those of corresponding thin-film stand-alone Si [2–6]. The PV efficiency calculated here ranges from 19% to 28% for AM1.5G solar irradiance [7] concentrated from 1 to ~ 1000 Suns. The result is for a flat uniform-planar cell, and its efficiency can be further increased via techniques such as surface passivation and light trapping treatments.

The solar device proposed here has technical and manufacturing issues. In this paper we investigate the former but not the latter. One technological issue is large-area low-cost manufacturing which usually includes the use of a low-cost “foreign” substrate such as glass or ceramic (preferably not Si). Another issue is defects at the hetero-interface that arise because of the Si/Ge lattice mismatch. Although technological issues of producing the Si/Ge heterostructure remain challenging, high-quality c-Si has already been grown successfully on c-Ge [8,9] with dislocations at the interface, together with some unwanted travel of Si atoms several nanometers into the Ge. Our goal is to point the way (with theory) to the development of practical high-efficiency cells.

2. Si/Ge tandem solar cell

Figure 1(a) shows the cross-sectional side view of our proposed thin-film c-Si on c-Ge PV cell whose overall thickness is less than ~100 μm. Since the Si has a wider bandgap than that of Ge, it is necessary in the Si/Ge tandem cell to have Si junction in front of the Ge junction. Both junctions have a 1-μm-thick “emitter” region with p-type doping of 5×10¹⁷ cm^-3. The “base” region thickness is 6 μm for the Si junction and is 90 μm for the Ge junction. Both bases have n-type doping of 10¹⁸ cm^-3. The tunnel diode at the Si-Ge interface has a thickness of ~100 nm with n-type Si and p-type Ge very heavily doped. The energy band diagram of this tandem cell is illustrated in Fig. 1(b) showing the carrier flow generated by solar irradiance.

Fig. 1 Illustration of (a) the Si-Ge tandem solar cell (b) the energy band diagram and carrier flow under solar irradiance.

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The predictions of the solar cell performance are made based on the following assumptions: (1) the well known experimental optical-and-infrared absorption spectra of Si and Ge [10] can be used here because the materials are mono-crystalline; (2) an anti-reflection (AR) coating at the front is employed, thus the efficiency figures cited do not include the 30% reflection loss at air-to-Si; (3) front-surface metal-finger contacts cover only a tiny fraction of the cell area , but perfect Ohmic contact at the back surface is assumed; (4) series resistance of the PV cell is neglected, but the resistive effect of the tunnel junction is included; (5) defects developed due to lattice mismatch is confined at the Si-Ge interface. The last assumption ensures that the defects will not significantly reduce the minority carrier lifetimes in the active regions of the solar cell. By confining the defects at the Si-Ge interface, their impact is limited only to the tunnel diode. One major effect of these defects is to introduce carrier-recombination centers within the tunnel diode that can serve to increase the tunneling current – a potentially positive impact on the current transport behavior.

3. Simulation method for the Si/Ge tandem cell

Let us consider that a specific solar spectral irradiance I(λ) (the solar intensity per unit wavelength interval at a given wavelength λ) is incident on the AR-coated front surface of the solar cell. In each of the solar junctions located in Si and Ge layers, the solar current is generated in three regions, two from the neutral regions and one from the depletion region, a current that can be determined by obtaining the distribution of excess minority carriers in those regions under solar irradiance. Within each of the neutral regions, the number of excess carriers generated at each wavelength λ per unit wavelength interval obeys the steady-state continuity equation with the approximation of zero electric field outside of the depletion region. We shall use the p-type neutral region of the Si layer $x_{0} < x < x_{1}$ to illustrate the procedure for obtaining the excess carriers in any of the four neutral regions of the two junctions. The steady-state continuity equation for $x_{0} < x < x_{1}$ can be written for excess electron concentration $Δ n_{λ} (x)$ under solar spectral irradiance I(λ) as

D_{e,} Si \frac{\partial^{2} Δ n_{λ} (x)}{\partial x^{2}} - \frac{Δ n_{λ} (x)}{τ_{e, Si}} + G_{λ} (x) = 0

where

D_{e, Si}

and

τ_{e, Si}

are the diffusion coefficient and lifetime of minority electrons, respectively. Similar continuity equations exist for excess carriers in the other three neutral regions. The carrier generation rate per unit wavelength interval in each region is

G_{i, λ} (x) = α_{i} (λ) F_{i - 1} (λ) \exp [- α_{i} (λ) (x - x_{i - 1})], x_{i - 1} < x < x_{i}

where the absorption coefficient

α_{i} (λ)

at wavelength λ within region i is either

α_{Si} (λ)

or

α_{Ge} (λ)

available from Palik [10] and the flux of photons per unit wavelength interval at the boundary of each region (

x = x_{i}

) can be obtained as

F_{i} (λ) = F_{i - 1} (λ) \exp [- α_{i} (λ) (x_{i} - x_{i - 1})]

Under the assumption of AR coating, the flux incident at the front surface (

x = x_{0}

) is given by

F_{0} (λ) = \frac{λ}{h c} I (λ)

where h is the Planck constant and c is the speed of light in free space.

The general solution to Eq.(1) for the density of excess minority holes per unit wavelength interval in the emitter region ( $i = 1$ ) is:

\begin{array}{l} Δ n_{λ} (x) = A_{1} \exp (\frac{x - x_{0}}{L_{e, Si}}) + B_{1} \exp (- \frac{x - x_{0}}{L_{e, Si}}) \\ + \frac{α_{Si} (λ) F_{1} (λ) τ_{e, Si}}{1 - α_{Si}^{2} (λ) L_{e, Si}^{2}} \exp [- α_{Si} (λ) (x - x_{0})], x_{0} < x < x_{1} \end{array}

where the diffusion length of minority holes

L_{e, Si} = \sqrt{D_{e, Si} τ_{e, Si i}}

, and the constants

A_{1}

and

B_{1}

are determined by boundary conditions that are related to the surface or interface recombination velocities

S_{i}

at

x = x_{0}

,

{D_{e, Si} \frac{d Δ n_{λ} (x)}{d x} |}_{x = x_{i}} = S_{i} Δ n_{λ} (x_{i})

The photocurrent density per unit wavelength interval that is collected at the depletion edge

x = x_{1}

due to excess electrons generated by incident light at the wavelengthλ, is

\begin{array}{l} J_{e} (λ) = e D_{e, Si} {\frac{d Δ n_{λ} (x)}{d x} |}_{x = x_{1}} \\ = e \frac{D_{e, Si}}{L_{e, Si}} [A_{1} \exp (\frac{x_{1} - x_{0}}{L_{e, Si}}) - B_{1} \exp (- \frac{x_{1} - x_{0}}{L_{e, Si}}) - \frac{α_{Si}^{2} (λ) F_{1} (λ) τ_{e, Si} L_{e, Si}}{1 - α_{Si}^{2} (λ) L_{e, Si}^{2}}] \end{array}

and that at the other depletion edge

x = x_{2}

due to excess holes

Δ p_{λ} (x)

is

J_{h} (λ) = - e D_{h, Si} {\frac{d Δ p_{λ} (x)}{d x} |}_{x = x_{2}} = - e \frac{D_{h, Si}}{L_{h, Si}} [A_{2} - B_{2} - \frac{α_{Si}^{2} (λ) F_{2} (λ) τ_{h, Si} L_{h, Si}}{1 - α_{Si}^{2} (λ) L_{h, Si}^{2}}]

where e is the electron charge.

Since the electric field inside of the depletion region ( $x_{1} < x < x_{2}$ ) is high, we can assume that the photogenerated carriers are swept out of the depletion region and are collected before any recombination takes place. Thus the photocurrent density collected from the depletion region is

J_{d} (λ) = e [F_{2} (λ) - F_{1} (λ)] .

The total photocurrent density for the top Si cell should then be integrated across the entire solar spectrum as

J_{photo} = \int [J_{h} (λ) + J_{e} (λ) + J_{d} (λ)] d λ .

Neglecting the series resistance from Ohmic loss and the shunt resistance from leakage currents, we can calculate the net current density of the PV cell under the operating voltage $V_{Si}$ as

J = J_{photo} - J_{s} [\exp (\frac{e V_{Si}}{k T}) - 1]

where J is the reverse saturation current density of the Si junction with the junction voltage

V_{Si}

. A similar expression can be obtained for the bottom Ge junction. We first design a structure that satisfies the current-matching condition at zero bias under solar irradiance by varying the thicknesses of Si and Ge layers as well as junction depths in them. The structure described above in Fig.1 provides such current matching. At each bias, the voltage is divided between three regions, the Si-junction

V_{Si}

, the Si-Ge tunnel junction

V_{t}

, and the Ge-junction

V_{Ge}

as

V = V_{Si} + V_{Ge} - V_{t} .

The voltages $V_{Si}$ and $V_{Si}$ are such that the currents flowing through both junctions are equal. The tunnel junction voltage $V_{t}$ is the bias necessary to pass the solar currents generated in the two junctions; it reduces the total voltage across the tandem PV cell, and depends on the passing current. The negative impact of $V_{t}$ can be neglected during “standard” irradiance of the tandem cell or when the solar concentration is low, in which case the tunnel junction can be treated as a perfect conductor between the two junctions ( $g = \infty$ , $V_{t} = 0$ ). But $V_{t}$ ’s impact is obviously more severe under high concentration of solar irradiance and should be included in the simulation. The tunnel junction in this cell is expected to work in the forward direction with its voltage below the peak voltage $V_{p}$ at which the tunneling current reaches its first peak $I_{p}$ before it enters into the negative resistance region. In this region, the tunneling current density can be approximated by a linear relationship $J_{t} = g V_{t}$ characterized by the conductance per unit area g whose effect is basically to increase the series resistance of the solar cell. The conductance parameter for the hetero Si-Ge tunnel junction is unknown and is currently being studied. A series of experimental samples formed by bonding a heavily doped n+ Si membrane onto a heavily doped p+ Ge substrate have yielded high-quality structures with defects confined at the Si-Ge interface. Tunneling behavior of the bonded Si/Ge is emerging in electrical characterization [11]. It is therefore feasible to produce Si-Ge tandem PV cells with top and bottom junctions free of misfit dislocations even though Si and Ge have a large lattice mismatch. Defects confined at the Si-Ge interface will not degrade the performance of either junction, and under forward bias these defects will serve as recombination centers for excess carriers that will increase the tunneling current. That increase can increase g – a potential advantage for the solar cell because it reduces the voltage drop across the tunnel junction. We have included such an effect in the simulation by considering a range of $g > 100 / Ω {cm}^{2}$ - a value well exceeded by the Ge tunnel diode [12].

Other parameters that have been used in the calculation are as follows. The Si and Ge absorption data are taken from Palik [10], and the diffusion coefficients for electrons and holes are taken as properties of elemental Si and Ge [13]; these are respectively, $D_{e, Si} = 39.0$ cm²/s, $D_{h, Si} = 11.7$ cm²/s, $D_{e, Ge} = 101.4$ cm²/s, and $D_{h, Ge} = 49.4$ cm²/s. All minority carrier lifetimes are assumed to remain at 1 μs for the range of solar power under consideration. The surface recombination velocities are 10⁴ cm/s at the front side where the metal fingers are used for contact, and ∞ for the backside where a metal sheet is used for contact.

4. Results and discussion

We have calculated the current-voltage (I-V) characteristic and efficiency of the PV cell using the above parameters. The results are shown in Fig. 2 for the AM1.5G solar irradiance under 500 Suns that is chosen to show the impact of voltage across the tunnel junction. While there are no differences in the short-circuit current I_sc and open-circuit voltage V_oc, the fill factor reduces to $F F = 0.74$ with $g = 100 / Ω {cm}^{2}$ from 0.83 when such a voltage is neglected ( $V_{t} = 0$ ). The short-circuit current and open-circuit voltage extracted from I-V characteristics are shown for Sun concentrations ranging from 1 to 1000 in Fig.3 where I_sc follows the solar power linearly; V_oc increases more rapidly from 0.9 V to 1.13 V in the 1 to 100 sun range, then slowly saturates as the concentration further increases. Both quantities show little dependence on the intensity of solar irradiance. But the fill factors are quite different as shown in Fig.4 for a range of the unit-area conductance g of the tunnel junction. When the series resistance of the PV cell is neglected, the fill factor increases slightly with the Sun concentration if the voltage across the tunnel junction is ignored. However, such a trend can be reversed when the limited conductance of the tunnel junction is taken into account. But for as long as the tunnel junction maintains a reasonable conductance of $g > 100 / Ω {cm}^{2}$ , the fill factor remains roughly unchanged throughout the 1-1000 range of suns concentration and is consistently above 0.8.

Fig. 2 I-V characteristics at 500 suns of AM 1.5G solar irradiance for $V_{t} = 0$ and $g = 100 / Ω {cm}^{2}$ .

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Fig. 3 Open-circuit voltage and short-circuit current vs. number of suns concentration.

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Fig. 4 Fill factor of the tandem PV cell vs. number of suns concentration for a range of the unit-area conductance of the tunnel junction.

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Figure 5 shows our main result – maximum efficiency as a function of AM 1.5G solar-irradiance concentration for the c-Si on c-Ge tandem solar cell depicted in Fig. 1. For comparison, we have also simulated a 100-μm-thick all-Si single-junction solar cell with the same doping profile as the Si region in the tandem cell. The Si cell has a 1-μm-thick “emitter” region and a 99-μm “base” region. The all-Si maximum efficiency is also shown in Fig. 5. Clearly, the tandem cell consistently delivers higher efficiency than the Si single-junction cell throughout the 1-1000 range of the suns concentration even for $g = 100 / Ω {cm}^{2}$ which can be easily exceeded by a reasonably good tunnel junction. For a modest tunnel junction with $g = 1000 / Ω {cm}^{2}$ , the efficiency increases from 19% to 28% as the sun concentration increases from 1 to 1000. In comparison with the all-Si single-junction solar cell, Si/Ge is 14% better at 1 sun, and continues to improve with increasing concentration: the Si/Ge improvement exceeds 20% at 50 suns, and reaches 23% at 200 suns and beyond.

The fabrication of c-Si on c-Ge solar cells faces challenges. The Si/Ge solar cells are expected to have defects due to the large lattice mismatch between Si and Ge. The defect density will depend on the method of cell preparation such as bonding a thin Si membrane to Ge or depositing Si on Ge via CVD. These defects can be localized at the Si/Ge interface or will exist in the form of threading dislocations. The former may not necessarily degrade the performance of the solar cell since the effect of defects only impacts the tunnel diode at the Si-Ge interface where defects serve as carrier-recombination centers that facilitate the tunneling current. However, threading dislocations will degrade the cell performance by creating additional recombination centers that decrease the lifetime of minority carriers both inside and outside of the pn-junction depletion region, resulting in smaller photo-current and larger dark current. It is therefore important to develop a method of fabrication that can effectively confine defects at the Si-Ge interface without allowing them to propagate into the active regions of either the Si or Ge layers.

Fig. 5. Efficiency of c-Si on c-Ge tandem PV cell vs. number of suns concentration for a range of the unit-area conductance of the tunnel junction. Also shown is the efficiency of an all-Si single junction PV cell of the same total thickness for comparison.

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5. Summary

In conclusion, we have proposed and analyzed a thin-film c-Si on c-Ge tandem PV cell. The structure is ~100-μm thick. Thicknesses of the various regions including junction depth are chosen so that the photocurrent-matching condition is satisfied. We have simulated tandem performance during standard AM1.5G solar irradiance with solar intensity ranging from 1 to 1000 suns. A general finding is that efficiency is optimized when the overall film thickness consists mainly of elemental Ge. We calculated efficiency for flat planar cells, assuming an AR coating on the front surface and no light-trapping enhancements such as texturizing or corrugating of the cell surfaces. We found that while the short-circuit current increases linearly with solar power, the open-circuit voltage increases super-linearly from 1 to 100 suns and then slowly saturates. The efficiency increases from 19% to 28% as the sun concentration increases from 1 to 1000. As a benchmark, we also calculated the efficiency of a 100-μm all-Si PV cell having the same parameters as the Si layer in the tandem structure. In comparison, the tandem c-Si on c-Ge PV cell delivers 14% more efficiency during standard AM1.5G solar irradiance. That improvement grows to 23% at 200 suns and beyond.

In this paper, we did not analyze and discuss the manufacturability of this Si/Ge cell, but we do recognize that economic and materials-science challenges must be met in order to realize the theoretical efficiencies in practical, large-area, cost-effective solar panels. The challenges are: attaining crystallinity, confining lattice-mismatch-induced defects at the Si-Ge interface, passivating the front surface, creating AR coating at the front surface, and doing all of the above at low cost including the Ge material costs. Looking to future experimental investigation of Si/Ge PV modules for solar panels, if it turns out after developmental effort that it will be too expensive to produce large-area solar panels, there is an excellent alternative “small-area approach” in which a large-scale array of small-area Si/Ge cells is deployed at the focus of a large solar-concentrator array. Also, it should be quite feasible to manufacture small-area Si/Ge PV cells for a host of “high value” applications in space platforms, airplanes and man-portable platforms.

To obtain group-IV performance that is improved over Si/Ge, the tandem’s Si-junction layer can be replaced by a group-IV alloy whose bandgap is wider than that of Si. For example, the top junction could be cubic germanium carbide with a bandgap of approximately 1.8 eV. According to detailed-balance theory, this 3C GeC-upon-Ge PV would have efficiency of more than 30%.

Acknowledgements

The authors are grateful to the Air Force Office of Scientific Research for partial support of this work.

References and links

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High efficiency thin-film crystalline Si/Ge tandem solar cell

Abstract

1. Introduction

2. Si/Ge tandem solar cell

3. Simulation method for the Si/Ge tandem cell

4. Results and discussion

5. Summary

Acknowledgements

References and links

Cited By

Figures (5)

Equations (12)

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