1 Introduction

Solar cells can be classified into first, second, and third generation. The first generation of solar cells consists of conventional solar cells, composed of P–N junctions of crystalline silicon (c-Si) wafers. The silicon used in these solar cells can be monocrystlyline (Mono-c Si) or polycristalline (Poly-c Si). There are also solar cells made of amorphous silicon (a-Si), but in this case, these solar cells may belong to the first or second generation.

Mono-c Si solar cells are obtained from a single Si crystal having a single orientation in their molecular structure and providing higher photovoltaic conversion efficiency (up to 26% [1]). On the other hand, Poly-c Si cells are made from several different Si crystals, i.e., they have heterogeneity in their internal molecular orientation (conversion efficiency up to 21% [1]).

The second generation of solar cells are known as thin-film solar cells (TFSCs), due to their layers of only a few micrometers thick. These solar cells consist of cadmium telluride (CdTe) and copper indium gallium selenide (CIGS) thin films (among others) and currently may have up to 12% conversion efficiency [1]. Although TFSCs have lower conversion efficiency, larger dimensions, and shorter longevity than c-Si solar cells, they are cheaper.

The third generation of solar cells, which may contain graphene, and/or other 2D materials, is also based on thin film but is still in the research and development stage. Examples of the third generation of solar cells are dye-sensitized solar cells (DSSCs) [2], organic solar cells (OSCs), or plastic solar cells, Perovskite solar cells [3], plasmonic solar cells, quantum dot solar cells, Schottky junction graphene/silicon solar cells [4,5,6,7,8,9], and graphene/other 2D materials hybrid solar cells.

Indeed graphene may be used in almost all cell types mentioned above. It is noteworthy that Indium Tin Oxide (ITO) is used with a non-conductive glass layer as a transparent electrode. However, as ITO is rare and brittle, the solar cells utilizing this material are expensive and unstable. Hence, as graphene has excellent conductivity, high transparency, low reflection, low electrical resistance, and cost-effective, it can replace ITO as a high-efficient transparent electrode for solar cells. Moreover, graphene has a bias tunable Schottky barrier height, via chemical or electrical doping, for example, enabling the replacement of the metal by graphene (with many advantages) in solar cells constituted by Schottky junction.

The power conversion efficiency (PCE) for graphene-based solar cells rapidly passed from the initial value of ≈ 1.5 to ≈ 15% [4,5,6,7,8,9], by controlling the number of graphene layers, addition of antireflection film, use of modified graphene films and silicon columns forming the Schottky junction, removing of PMMA residues left in the graphene during its transfer process, as much as possible, and controlling of the graphene work function by the insertion of chemical doping. However, the large loss of the photogenerated due to the current that crosses the Schottky junction (Id), and does not contribute to external current, was not considered. On the other hand, in the solar cell we are presenting, there is practically no Id current so that all photogenerated current goes to the external circuit, and the power conversion efficiency (PCE) of this solar cell is still enormously increased. This is possible by the use of more appropriate values of the Schottky junction parameters. Note that the increase of the Schottky barrier height (SBH) can be obtained by changing the graphene work function. Hence, as a higher SBH causes a great decrease of the saturated reverse current (I0), and consequently, for a very small I0 (as it has been reached), the current that crosses the diode (for the voltage range of this type of solar cell) is very small so that it can be neglected.

Currently, the maximum SBH for Schottky junctions consisting of p-doped graphene/n-doped silicon (p-G/n-Si) is ≈ 0.8 eV. On the other hand, an n-doped graphene/p-doped silicon (n-G/p-Si) Schottky junction can be obtained from graphene doped with metallic dopants so that its SBH has a very higher value (≈ 2.94 eV). Therefore, as the n-G/p-Si Schottky junction has an SBH and built-in potential (Φbi) with very high values, solar cells constituted by this referred Schotty junction have a high power conversion efficiency.

We are presenting a solar cell based on the n-G/psi Schottky junction, working similar to an electrochemical generator that has a very high energy conversion efficiency, which may exceed the Shockley–Quiesser limit.

This manuscript has three more sections. In Sect. 2, we detail the starting point for the determination of all parameters of the n-G/p-Si solar cell we are presenting. In Sect. 3, we present details of that referred n-G/p-Si solar cell. Section 4 is devoted to the conclusions regarding what we are presenting.

2 p-Graphene/n-silicon and n-graphene/p-silicon solar cells (start point)

In Fig. 1, we are presenting a simple but efficient way to determine the operation of a graphene/silicon Schottky junction solar cell.

Fig. 1
figure 1

Schematic representation of the operation of a Schottky junction-based p-G/n-Si solar cell. Upper left: external circuit with resistance R. Upper right: short circuit. Bottom left: open circuit. Bottom right: electrical circuit

Full size image

As it is shown in the upper left-hand part of this Figure, the generated charges (due to the absorption) are separated by the built-in field so that in the p-graphene (p-G)/n-silicon (n-Si) solar cells, holes are routed to the graphene, and electrons are moved to the silicon. However, these charges provide a forward bias polarization of the Schottky junction so that part of the photogenerated charges cause an internal current through the forward bias Schottky junction (diode current—Id). Hence, only a part of the generated charges contributes to the external current (I, electrons current, first quadrant).

Since the value of the shunt resistance (Rsh) related to this type of solar cell is very high (up to 45 MΩ) [6], we can neglect the leakage current Ish. Therefore,

$$I = I_{ph} - I_{d} \to I_{d} = I_{ph} - I$$
(1)

where \(I_{ph} \approx I_{sc}\) represent the photogenerated current and the short-circuit current, respectively.

In the upper right-hand part of Fig. 1, we are showing the behavior of this solar cell operating with an external short circuit. Note that in this case, we can consider that the voltage applied at the Schottky junction is zero, which causes the current Id not to exist. Hence, in this case, all the photogenerated charges contribute to the external current (whose value is Isc—short circuit current). On the other hand, in the lower left-hand part of Fig. 1, it is shown the open circuit operation, where we can observe that the voltage applied at the Schottky junction has the maximum value (Voc—open circuit voltage). Therefore in this situation, the current through the diode has maximum value (Isc). We can see the schematic representation of the electric circuit of this type of solar cell in the lower right-hand part of Fig. 1.

Although the process of cleaning, transfer, and doping of graphene, has been greatly improved, the power conversion efficiency (PCE) of this type of solar cell is still low. This mainly occurs, because the electric current flowing through the forward-biased diode is relatively high, and does not contribute to the external circuit current. In other words, according to Eq. 1, the value of the external current decreases as a function of the current that crosses the forward bias diode.

To facilitate the presentation of the n-G/p-Si solar cell we are presenting, we will first detail a p-G/n-Si solar cell.

Let us assume the data presented for a given p-G/n-Si solar cell [1]: (V, I) measured data, open circuit voltage Voc ≈ 0.548 V, and short circuit current Isc ≈ 7.819 mA (solar cell exposed to the environment). We got the (V, I) curve (and (V, Id) curve referring to the solar cell mentioned above, using several points of the (V, I) curve obtained via simulations, according to the which is shown at the top of Fig. 2 (solid and dashed line, respectively; y-axis located on the left side).

Fig. 2
figure 2

Top: (V, I) curve (V, Id) curve, (y axis located on the left side), and (V, P) curve (y axis located on the right side), referring to the p-graphene/n-silicon solar cell mentioned above. Bottom: (I, V) curve

Full size image

In this same Figure, we are showing the (V, P) curve (dashed and dotted line; y-axis located on the right side). The value of the maximum power is Pmax ≈ 0.00222 W, the fill factor value is \(FF = V_{\hbox{max} } *I_{\hbox{max} } /V_{oc} *I_{sc} \approx 51.69\%\), and the PCE value is ≈ 11.66%.

We determined the PCE value using the following Equation:

$$\eta = \frac{{P_{m} }}{AG}$$
(2)

where Pm is the maximum power supplied by the solar cell and G = 1000 W/m2 is the value of the solar radiation power under the AM1.5 conditions.

There are several analytical methods for determining the parameters of a solar cell from the data obtained through simulations [10,11,12,13,14]. However, to show the behavior of the p-G/n-Si solar cell, we used the data obtained in the above-mentioned simulations.

At the bottom of Fig. 2, we are showing the graph (I, V). Note that the voltage at the external circuit varies according to the value of the external current intensity. Moreover, the voltage value in the external circuit decays approximately linearly as a function of the current rise up to 5 mA. From this value, it decays exponentially and has a null value in the short-circuit condition.

Due to its imperfections and the charged impurities located on the substrate, real graphene has non-zero intrinsic charge density [15, 16]. Hence, its Fermi level is not located at the Dirac point. In a research carried out it has been demonstrated that the dry transfer of graphene films to the standardized substrate (SiO2/Si), as well as the subsequent cleaning through modified Radio Corporation of America (RCA) procedure, provides production of devices composed of graphene with low Dirac voltage (VD < 4.5 V) [17]. In a research where the quantum capacitance of graphene has been measured (single and bilayer samples) as a function of gate potential, it has been found the residual graphene carrier concentration n* ≈ 8 × 1011 cm−2 [18]. Note that the Dirac voltage value depends on the efficiency of the graphene manufacturing and transport process.

Changes in graphene work function values, due to the effects of doping of graphene with arsenic and with phosphorus were elaborated through a Metal–Organic Chemical Vapor Deposition (MOCVD) reactor. Therefore, using the chemical doping of graphene, with phosphor and arsenic, its work function increases from 4.5 eV (undoped graphene) to 4.7 eV and 4.8 eV, for phosphor and arsenic, respectively, and as a consequence, there is an increase of the potential barrier [19]. However, this value of \(\varPhi_{B}\) is still very low, so that the current through the forward-biased diode is very high.

Taking into consideration that in the n-G/p-Si solar cell the value of \(\varPhi_{B}\) is relatively high (this subject will be detailed later), for comparison, we will determine the potential barrier value for a Schottky p-G/n- Si, with graphene having work function \(\varPhi_{G} = 4.8\) eV.

It is well known that for a Schottky junction, the Fermi level of the n-silicon substrate (after the depletion zone) is shifted upwards, according to the following Equation:

$$\Delta E_{F} = K_{B} Tln\left( {\frac{N}{{n_{i} }}} \right).$$
(3)

As the charge doping density of n-Si is \(N = 10^{17} \;{\text{cm}}^{ - 3}\), the silicon intrinsic density of charge carriers \(n_{i} \approx 1.45 \times 10^{10} \;{\text{cm}}^{ - 3}\) (value found from the effective densities of states of electrons and holes in the conduction and valence bands, respectively, of a silicon intrinsic semiconductor), the value of the displacement of the silicon Fermi level is \(\Delta E_{F} = 0.407\;{\text{eV}}\) (upward). Therefore, in this case, the energy level of the n-silicon conduction band is given by \(E_{c} = E_{g} /2{-}\Delta E_{F} \approx 0.153\;{\text{eV}}\) (Eg = 1.12 eV is the silicon bandgap value) above the Fermi level. Hence, considering that the value of the electron affinity of silicon is \(\chi = 4.05\;{\text{eV}}\), and \(\varPhi_{G}\) = 4.80 eV we determined, analytically, the values of the Schottky barrier height (\(\varPhi_{B} = \varPhi_{G} - \chi \approx 0.75\;{\text{eV}}\)) and of the built-in barrier height (\(\varPhi_{bi} = \varPhi_{G} - \chi - E_{c} \approx 0.597\;{\text{eV}}\)) referring to this p-G/n-Si Schottky junction, as shown in Fig. 3 (without illumination).

Fig. 3
figure 3

p-G/n-Si Schottky junction energy band structure at thermodynamic equilibrium (without illumination)

Full size image

It is noteworthy that the \(\varPhi_{B}\) value determined by this methodology is very close to the actual value since only the influences concerning the image force barrier lowering and charge traps on the interface and native oxide are not considered. However, the image force barrier lowering produces only small deviations from the value of \(\varPhi_{B}\) [20]. Furthermore, by using an optimized graphene transfer process, it is possible to obtain an interface with a small number of atomic defects. This procedure, together with actions that prevent the formation of silicon dioxide, provides the reduction of the charge traps so that we can neglect the influence of these charge traps [21]. It is noteworthy that the determination of the Schottky barrier height is the starting point for our n-G/p-Si solar cell.

From thermionic emission theory, the Equation that provides the current through the external circuit (I) in a Schottky junction solar cell as a function of the voltage (V) (in the load resistance) is given by:

$${\text{I}}\left( {{\text{T}},{\text{V}}} \right) = {\text{I}}_{0} \left[ {{\text{e}}^{{\frac{{{\text{qV}}_{d} }}{{\upeta{\text{K}}_{\text{B}} {\text{T}}}}}} - 1} \right] + \frac{{{\text{V}} + {\text{R}}_{\text{s}} {\text{I}}}}{{{\text{R}}_{\text{sh}} }} - {\text{I}}_{\text{ph}} ;$$
(4)
$$I_{0} \left( T \right) = AA^{*} T^{2} e^{{ - \frac{{q\varPhi_{B} }}{{K_{B} T}}}} ,$$
(5)

which can be solved numerically. We used MATLAB for the numerical resolution of all Equations presented in this manuscript.

It is worth mentioning that in Eqs. 3 and 4, I0 is the saturated reverse current, Vd is the voltage at the Schottky junction, η the ideality factor, KB the Boltzmann constant, T the ambient temperature, A the effective area of contact, and A* the Richardson’s constant.

Entering the value of ΦB given above in Eq. 5 (considering A = 0.19 cm2), we determined the value of I0 = 4.822 × 10−7 A, which is still very high.

It is important to state, that the greater the value of \(\varPhi_{B}\), the lower the value of I0, and consequently, the lower is the Id value. So we must choose a solar cell structure that gives a large value of \(\varPhi_{B}\). Therefore, the doping of graphene is essential to increase the efficiency of this type of solar cell.

3 n-Graphene/p-silicon-based Schottky junction solar cell with very high PCE

According to we have mentioned earlier, larger SBH provides smaller saturated reverse current value, and consequently, smaller current crossing the Schottky junction operating as a forward-biased diode. From the energy band diagram, related to the solar cell to be designed, we can obtain the highest possible SBH, by choosing the more appropriated types of doping for graphene and silicon (n, or p).

n-doped graphene (n-G) can be obtained from metallic dopants so that its work function is reduced (for example, to 2.23 eV, using K-doped graphene [22, 23]). Therefore, an n-G/p-Si Schottky junction may have an SBH with a much higher value than a p-G/n-Si Schottky junction, as we can see in Fig. 4.

Fig. 4
figure 4

n-G/p-Si Schottky junction energy band structure at thermodynamic equilibrium (without illumination)

Full size image

For the elaboration of Fig. 4, we used the same values related to charge doping density (\(N = 10^{17} \;{\text{cm}}^{ - 3}\)), as well as intrinsic density of charge carriers (\(n_{i} \approx 1.45 \times 10^{10} \;{\text{cm}}^{ - 3}\)), and electron affinity for the silicon used in the p-G/n-Si Schottky junction. As we can see, we determined analytically the values of the Schottky barrier height (\(\varPhi_{B} = E_{g} + \chi - \varPhi_{G} \approx 2.94\;{\text{eV}}\)), and of the built-in barrier height (\(\varPhi_{bi} = \varPhi_{B} {-}E_{v} \approx 2.787\;{\text{eV}}\), where \(E_{v} = E_{g} - E_{c}\) is the energy level of the p-silicon valence band), referring to the n-G/p-Si solar cell we are presenting.

It is noteworthy that we can use the following equation to determine the value of the variation of the Fermi level (or the chemical potential) of graphene, \(\Delta \mu_{g}\)) [24, 25]:

$$\Delta \mu_{g} = \Delta E_{F} = \hbar {\text{V}}_{\text{F}} \sqrt {{\text{n}}\uppi} ,$$
(6)

where \(\hbar\) is the Planck’s reduced constant, VF is the Fermi velocity for graphene (106 m/s), and n the charge carrier density. So, as the graphene Fermi level can be altered in several ways (for example, via chemical doping, or gate voltage), the graphene work function can also be changed, and consequently, the graphene/semiconductor potential barrier can be altered too, according to what was mentioned previously.

On the other hand, due to the application of an external voltage, there is a variation of the density of electrons located in graphene (ng) and holes in n-Si at a graphene/silicon Schottky junction. Note that the variation of the density of electrons in graphene is the same variation of the density of holes located in the n-semiconductor depletion region [26]. For forward voltage, the number of holes in the semiconductor depletion zone decreases, which leads to the decrease of electrons in the graphene. So, in this case, the Fermi level is shifted down. It is noteworthy that for reverse voltage (considered as V < 0), the number of holes in the depletion zone increases, resulting in the negative value of ng, so that the Fermi level of the graphene is shifted upwards. Therefore, with the Schottky junction under illumination, the Fermi level of graphene is shifted downward (forward bias diode). However, due to the small values of the voltage in a solar cell, this displacement is very small and can be neglected.

By inserting the value of the SBH in Eq. 5, we determined the value of I0 ≈ 7.84 × 10−44 A. Hence, in this case, the current flowing through the forward-biased diode is very small, so that it can be neglected, according to what we show below.

In Fig. 5 we are showing the schematic representation of the n-G/p-Si solar cell we are presenting.

Fig. 5
figure 5

Schematic representation of the operation of a solar cell based on n-G/p-Si Schottky junction

Full size image

Since there is no current flowing through the Schottky junction, in this case, all the photogenerated charge is routed to the electrodes. Hence, in this type of solar cell, the external current is provided by the fixed voltage that occurs in the Schottky junction (electromotive force (ε)), which is given by ε = V + IRs, as we can see from Fig. 5.

To represent the behavior of the n-G/p-Si solar cell, we first consider the voltage at the Schottky junction referring to the p-G/n-Si solar cell, which is given by Vd = V + IRs.

We obtained the Rs value (17.52 Ω) through the methodology presented in Ref. [27], taking into account the range of values close to the final part of the (V, Id) curve referring to the forward bias diode (p-G/n-Si solar cell). It is worth mentioning that the series resistance of this type of solar cells presented values between 9 and 12 Ω, for a voltage range between 0.1 and 2.0 V [6].

The curves (I, V) and (I, Vd) are shown at the top of Fig. 6 (for Rs = 17.52 Ω).

Fig. 6
figure 6

Top: (I, V), (I, Vd), and (I, ε) curves. Bottom: (V, I) curves for n-G/p-Si (solid line) and p-G/n-Si (dashed line) solar cells

Full size image

Note that the value of Isc occurs for the minimum value of Vd (≈ 0.137 V). Also, note in this Fig. 6, the graphical representation of the constant value of ε for the n-G/P-Si solar cell.

On the other hand, as in the n-G/P-Si solar cell, there is no current flowing through the forward bias diode, all the current goes to the external circuit so that this solar cell we are presenting operates similarly to an electrochemical generator. Note that in this case, this solar cell has an electromotive force ε = max(V + IRs) = Voc. Therefore, in this case, Isc = ε/Rs ≈ 0.0313 A.

For comparison, the (V, I) curves for n-G/p-Si (solid line) and p-G/n-Si (dashed line) solar cells are plotted at the bottom of Fig. 6. Note that the curve (V, I) for the solar cell n-G/p-Si is a straight line, just as it is in an electrochemical generator.

The graphs (V, P) for p-G/n-Si and n-G/p-Si with Rs = 17.52 Ω, and Rs = 10.00 Ω solar cells are shown in Fig. 7.

Fig. 7
figure 7

(V, P) curves (p-G/n-Si, n-G/p-Si solar cell with Rs = 17.52 Ω, and n-G/p-Si solar cell with Rs = 10.00 Ω)

Full size image

We determined the value of the maximum power and PCE of the solar cells p-G/n-Si, and n-G/p-Si with Rs = 17.52 Ω, and Rs = 10.00 Ω i.e., Pmax ≈ 0.00222 W (Vmax ≈ 0.355 V, Imax ≈ 0.0063 A); PCE ≈ 11.66% (these values have already been shown in Sect. 2), Pmax ≈ 0.00429 W (Vmax ≈ 0.274 V, Imax ≈ 0.0157 A); PCE ≈ 22.55%, and Pmax ≈ 0.0075 W (Vmax ≈ 0.274 V, Imax ≈ 0.0274 A); PCE ≈ 39.51%, respectively.

As we can see, the value of Rs, which depends on the solar cell’s manufacturing process, is essential for obtaining solar cells with a high PCE value.

It is noteworthy that the PCE value for n-G/p/Si can exceed the value of the Shockley-Quiesser limit [28]. This is because in the Shockley-Quiesser limit (PCE ≈ 28% for silicon PN junction-based solar cells), it has not considered the use of the current through the forward-biased diode.

To prove that actually in the n-G/p-Si solar cell there is no electric current flowing through the Scottky junction, we plotted the (V, I) curve for a diode, considering \(\varPhi_{B}\) = 2.94 eV, Rs = 10.00 Ω, and neglecting a shunt resistance, as shown in Fig. 8.

Fig. 8
figure 8

(V, I) curves referring to the diode representing the Schottky junction of the solar cell p-G/n-Si (forward-bias) for η = 1.5 and η = 2.0

Full size image

Note that considering the ideality factor η = 1.5, only for voltage higher than 2.5 V, occurs a current across the diode, and for η = 2.0, only for higher voltage values than 3.5 V, occurs a current across the diode. Therefore, as the maximum voltage in the solar cell was considered 0.548 V, we can state that there is no current crossing the Schottky junction so that all photogenerated current goes to the external circuit.

Moreover, for p-G/n-Si solar cells with a small value of Rs, the external current increases (considering a fixed value of the external resistance (R)) so that the value of the external voltage also increases. Therefore, the PCE value of the solar cell p-G/n-Si increases too. However, in this case, the PCE value of this solar cell remains below the Shockley-Quiesser limit.

Since the generated photocurrent density (Jph) in a solar cell is given by [29]:

$${\text{J}}_{\text{ph}} = \mathop \int \limits_{{\lambda_{1} }}^{{\lambda_{2} }} {\text{qT}}\left(\uplambda \right)\varPhi_{\text{in}} \left(\uplambda \right)\left( {1 - {\text{e}}^{{ -\upalpha\left(\uplambda \right){\text{w}}}} } \right){\text{d}}\uplambda,$$
(7)

where λ1 and λ2 are the minima and maximum wavelengths, that penetrate the semiconductor (UV and the semiconductor cutoffs, respectively), T (λ) is the graphene transmittance, \(\varPhi_{in}\)(λ) is the incident photon flux, α(λ) is the optical absorption coefficient of the semiconductor and w is the width of the depletion zone, we can affirm that the type of doping and the \(\varPhi_{B}\) value do not influence the ideal generated photocurrent density.

We can further affirm that since the built-in barrier at the n-G/p-Si Schottky junction is much larger than at the p-G/n-Si Schottky junction, the number of recombinations that arise in this solar cell we are presenting should be smaller than in the p-G/n-Si solar cell. Therefore, it is expected that the n-G/p-Si solar cell has higher values of Voc, Iph, and PCE than those determined above (referring to the n-G/p-Si solar cell).

4 Conclusions

Currently, the maximum Schottky barrier height (SBH) for Schottky junctions composed of p-doped graphene/n-silicon is ≈ 0.8 eV. On the other hand, n-doped graphene can be obtained from metallic dopants, so that its SBH has a higher value. We presented a solar cell constituted by n-graphene/p-silicon Schottky junction, whose SBH is ≈ 2.94 eV. Due to this high SBH value, the saturated reverse current (I0) value is very low, so that the current flowing through the Schottky junction can be neglected, which leads to an increase in the PCE value.

We showed that considering the series resistance Rs = 17.52 Ω, and Rs = 10.00 Ω, the PCE values are ≈ 22.55%, and ≈ 39.51% (in this case, the PCE value is higher than the Shockley-Quiesser limit), respectively. Moreover, due to the great value of the built-in field in the n-G/p-Si solar cell we presented (which operates similarly to an electrochemical generator), the amount of recombination should be smaller than in the p-G/n-Si solar cell so that the PCE value of the n-G/p-Si solar cell can be even higher.

To show the characteristic parameters of this n-G/p-Si solar cell, we used an analytical/numerical methodology.