A p-Si/CoPc Hybrid Photodiode System for Looking at Frequency and Temperature Dependence on Dielectric Relaxation and AC Electrical Conductivity

The system response upon the application of an alternating signal through a specimen is described by CIS. The imaginary and real parts can be separated to produce a picture of the tested device properties. The rules of the complex impedance \({(Z}^{*})\) of the tested photodiode are obtained as follows:¹⁸where Z′ is the real part of the impedance, \(Z^{\prime\prime}\) is the imaginary part, \(Z\) is the detected impedance, and (\(\theta\)) is the phase angle. The dependence of frequency on Z′ at different temperatures is depicted in Fig. 2. Z′ shows that lower frequencies are associated with resistance of recombination (R_rec) at the p-Si/CoPc interface, the resistance of transport (R_tran) at the Al/p-Si interface, and series resistance (R_s). At higher frequencies, the values of the series resistance can be deduced at various temperatures.^11,17,19

A gradual decline can be observed for Z′ values at lower frequency values. In this region, impedance is as a reason of the interface and the bulk of this device.¹⁷ It can be observed that impedance declines as the applied frequency increases. Z′ is strongly affected by temperature at low frequencies, but is largely independent of temperature at higher frequencies. The decline in Z′ values as temperature increases is an indication of a negative temperature coefficient of resistance (NTCR) in the Al/p-Si/CoPc/Au structure. Figure 3 shows the changes to Z″ as a function of the applied frequency at various temperatures. In this figure, the values of Z″ increase as the frequency increases, reaching a maximum peak (Z″_max). Then, the values of Z″ decline as the applied frequency increases. The peaks broaden as the temperature increases, suggesting the spread of relaxation time (i.e., the existence of temperature is dependent on electrical relaxation). The values of Z″ decline as temperature increases at lower frequencies, but they merge for all temperatures at higher frequencies. The spectral distribution of Z″ values can be used to estimate the values of recombination resistance, where Z″_max = R_rec/2.^20‐24

The values of recombination resistance from the fitting Cole–Cole plots match and confirm the estimate from the spectral distribution of Z″ values. At a frequency ranging between 20 kHz and 200 kHz, a shoulder appears owing to the capacitive nature of the Al/p-Si interface.²³ At higher frequencies, the aggregation of space charges in the photodiode causes a merger of both Z′ and Z″ values.^9,13 A clear shift in the position of the peaks towards the high frequency limits is observed at divergent temperatures when the applied frequency increases. This influences the mobile charge carrier’s relaxation time (\(\tau\)), which declines as temperature increases. This can be evaluated using the following relation:where \(f_{\max }\) is the maximum frequency. The dependence of the time of relaxation on temperature is shown in Fig. 4. The relaxation time values decline as temperature increases. The electrical charges are thermally influenced and excited to certain levels. Consequently, the formed dipoles can follow the alternating field with the help of dissipated thermal energy,²⁵ making the time of relaxation shorter. On the interface of the p-Si/CoPc layers, the dissociation of the excitons and the rapid transport of charge carriers can be detected via the brevity of the relaxations.²⁶ The next relation indicates the time of relaxation:²⁷where \( \tau_{0 }\) is a constant characteristic of the time of relaxation expressing the period of a single oscillation of the dipole in the potential well: \(\Delta E_{\tau }\) is the activation free energy for the relaxation of dipoles. The inset in Fig. 4 shows the plot of \(\ln \left( \tau \right) \) versus 1000/T for the tested photodiode: \(\Delta E_{\tau }\) values are also estimated and recorded in Table I.

The relaxation of the dielectrics through the frequency response of the tested photo-junction is a crucial phenomenon that can be examined through Cole–Cole plots and their corresponding fits of the Al/p-Si/CoPc/Au structure at various temperatures: such is shown in Fig. 5a, where the imaginary impedance values are plotted against the real ones. At various temperatures and frequencies, each Nyquist plot indicates two semicircles, which affirms the presence of more than one relaxation mechanism (nearly two) at the two interfaces (Al/p-Si and p-Si/CoPc).

A temperature-dependent mechanism for relaxation can be found by looking at the semicircle diameters in the given impedance spectra, which rely strongly on temperature. By examining the diameters, it can be seen that the electrical conductivity of the tested device and the maximum values match the characteristic frequency of relaxation. The stability of the Al/p-Si/CoPc/Au structure arises from the presence of regular and uniform semicircles: the homogeneous electrical response probably derives from an electro-active limit.^27‐29 In Fig. 5b, the equivalent circuit used to model the impedance spectra is presented. To create this equivalent electric circuit in the Al/p-Si/CoPc/Au structure, two parallel circuits in series were connected to contact resistance in series. Specifically, one of these electric circuits involves resistance and capacitance, while the other one includes a resistance and constant phase element (CPE). The use of an imperfect capacitor, the CPE, in the equivalent electrical circuit is necessary because of the heterogeneous grain distribution and the slashed arcs or semicircles in the Cole–Cole plots. By using electrochemical impedance spectroscopy (EIS) and spectrum analyzer software³⁰ (adjusting some parameters and fitting certain data), the elements of the equivalent electric circuit were estimated and are recorded in Table I. To describe these components in the model equivalent circuit, R₁ stands for contact resistance (series resistance) and/or the electrode system (entire bottom region, Al, and on top, Au grid). The transfer and recombination operations of the charge carriers at the p-Si/CoPc interface (R₃, CPE) are characterized by a large semicircle within the low frequency limit, which corresponds to the equivalent circuit.^23‐25

Thus, R₃ denotes the shunt resistance connected to the semicircle’s diameter and the recombination rate through the direct and inverse relations, respectively. The space charge region capacitance (junction capacitance), a function of the photodiode response, is denoted by the CPE symbol in the equivalent circuit. At the high frequency limit, the recombination and charge transfer operations at the conducting Al/p-Si interface are described by the small semicircle (R₂, C₁).^26,27,29 R₂ is identical to the resistance of transport, and C₁ corresponds to the traps and imperfections at the Al/p-Si interface.²⁸ The behavior of R₂ at the Al/p-Si interface and R₃ at the p-Si/CoPc interface in terms of temperature can be described as follows:³⁰where R₀₂ and R₀₃ are pre-exponential factors, and ΔE_a2 and ΔE_a3 are the activation energy values at the two interfaces. Figures 6 and 7 illustrate the change in ln (R₂) and ln (R₃), respectively, as a function of 1000/T at the Al/p-Si and CoPc/p-Si interfaces.

At the (Al/p-Si) interfaces, two linear curves are noted; this agrees with the results provided by Huang et al.²³ The values of the activation energy can be estimated from the slope, as shown in Table I. The law to evaluate the real capacitance C₂ (junction capacitance) of a CPE can be written as follows:³¹where (\(n\)) is the CPE exponent with a value of (0 ≤ \(n\) ≤ 1). This characterizes the capacitive extent of the component, and an ideal capacitor has a range of 0.9–1. P is a pre-exponential factor.³¹

For the tested device, the Fermi level increases and decreases with respect to the levels at the semiconductor interface when the signal of the AC source is used to modify the charge of traps on the interface.²⁶ At higher frequency values, the levels at the interface (p-Si/CoPc) do not follow the AC high-frequency signal with sufficient rapidity; therefore, they do not participate in the detected capacitance.²⁶ At lower frequency values, the width of the depletion region (\(W\)) at the p-Si/CoPc interface can be deduced as a variation in the capacitance of junction C₂ as follows:^23,32where (\(A\)) is the Au cross-sectional grid area of the electrode and takes the value of 0.0001 m², \(\varepsilon_{0} = 8.854 \times 10^{ - 12} \frac{F}{m} \) is the permittivity of free space, and \(\varepsilon = 3.6 \) is the relative permittivity of the thin CoPc film.³³ For the tested device, the lifetime of the excess minority carriers at the Al/p-Si interface (\(\tau_{1}\)) and the lifetime of the excess minority carriers at the p-Si/CoPc (\(\tau_{2}\)) interface before recombination can be given from the following relations:³⁴

The Arrhenius law is used to illustrate and explain the dependence of the minority carrier lifetime at both interfaces as follows:where \( \left( {\tau_{01} } \right) \) and \(\left( {\tau_{02} } \right) \) are constants of relaxation time and (\(\Delta E_{\tau 1}\)) and (\(\Delta E_{\tau 2}\)) are the activation energies at the two interfaces. Figures 8 and 9 show the variations of ln \(\left( {\tau_{1} } \right)\) and ln \( \left( {\tau_{2} } \right)\) versus 1000/T at the two interfaces, respectively, while the slopes of the linear fit in Figs. 7 and 8 give the activation energy values.

For the tested heterojunction, the coefficients of diffusion and the mobilities at the Al/p-Si interface (\(D_{1} ,\mu_{1} )\) and at the Al/p-Si CoPc/p-Si interface (\(D_{2} ,\mu_{2}\)) can be calculated as follows:³²where \(d\) is the thickness of the interfacial organic layer, \(q\) is the charge of the electric charge carriers, and \(T\) is the absolute temperature.

Table II indicates the measured and calculated values, which are \(C_{2} ,W,\tau_{1}\),\( \tau_{2}\),\( D_{1}\),\( D_{2}\),\( \mu_{1} , \) and \( \mu_{2}\). The charge mobility is highly dependent on temperature, so its behavior as a function of temperature is examined in this work.

The dependence of the diffusion coefficient \(D_{1}\) and carrier mobility \(\mu_{1}\) at the Al/p-Si interface (and also \(D_{2}\), \(\mu_{2} \) at the p-Si/CoPc interface) have been examined in terms of the Arrhenius law:²³

The pre-exponential factors (\(D_{01}\), \(D_{02}\), \(\mu_{01} ,\) and \(\mu_{02}\)) are constants of the diffusion coefficient and mobility values, respectively, and\(\left( {\Delta E_{D1} , \Delta E_{D2} ,\;\Delta E_{\mu 1} ,\;{\text{and}}\;\Delta E_{\mu 2} } \right)\) are the corresponding values of activation energy formed at the two interfaces Al/p-Si and p-Si/CoPc, respectively. The behaviors of (ln \( D_{1} \)–ln \( \mu_{1}\)) and (ln \( D_{2 }\)–ln \( \mu_{2}\)) versus (1000/T) at both interfaces are depicted in Figs. 10, 11, 12 and 13, respectively; their activation energy values are determined from the slopes of the fitting lines.

The best description of electrical behavior can be obtained through studying the dielectrics in organic matter as a variation of frequency and temperature. This complex dielectric relationship is expressed as follows:³⁵where \(\varepsilon^{\prime}\) is a constant concerning the dielectric substance and is connected to the matter’s capacitive nature, providing a measurement of the reversible energy stored by polarization inside the device.³⁵ \(C\) is the capacitance and \(d\) is the thickness of the interfacial organic layer.³⁶ \(\varepsilon^{\prime\prime}\) is the loss due to dielectrics and is a measure of the energy needed for molecular motion.³⁵ Figures 14 and 15 illustrate the dependence of frequency on the dielectric constant and the dielectric loss, respectively. These graphs show the declining values of the real and imaginary parts of the dielectric as the frequency increases; this is due to dielectric polarization. At lower frequencies, the high values of the two dielectric terms tend towards the space charge; interfacial polarization arises from the charge accumulated at the p-Si/CoPc interface. At the low frequency limit, this is explained by the free charges at the interfaces inside the bulk of the specimen (interfacial \({\text{Maxwell}}\)–\({\text{Wagner}}\) (\({\text{MW}}\)) polarization) and at the interface between the specimen and the electrodes (space charge polarization).³⁷ When the frequency increases, charge agglomeration cannot take place due to the time required for the process to take place, but this can only happen at the interfaces.^38,39 As the frequency increases, interfacial polarization decreases, which in turn causes a decline in the dielectric terms (\(\varepsilon^{\prime},\varepsilon^{\prime\prime}\)). On the other hand, as temperature increases up to 353 K, the real dielectric constant term decreases. This phenomenon is explained by space charge polarization and the reversal of the direction of polarization.^40,41 The dielectric constant increases above 353 K due to the increase in temperature, which stimulates molecule expansion: this leads to an increase in electrical polarization and an increase in the dielectric constant. The increase in \(\varepsilon^{\prime\prime}\) against temperature can be shown as follows. The relaxation processes are divided into three items: conduction loss, dipole loss, and vibrational loss. As the temperature increases, conduction loss increases, causing an increase in \(\varepsilon^{\prime\prime}\).^7,42 Additionally, at higher temperatures, the dipoles reach complete rotational freedom, and the effect of molecular interaction energy becomes lower than that of thermal energy; this causes an increase in \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\).⁴³

The dependence of dielectric loss (\(\varepsilon^{\prime\prime}\)) on frequency (\(\omega\)) is indicated by the following formula:where \(B\) is a temperature-related constant, ω is the angular frequency of the applied AC field, and \(m\) is the exponential factor. Figure 16 shows the dielectric loss variation as a function of the applied frequency; the values of \(m\) can also be obtained. The two linear parts at the CoPc/p-Si and Al/p-Si interfaces are situated at low and medium frequencies, respectively.

The following formula shows the relationship between the exponential factor (\(m\)), temperature \(\left( T \right)\), and the maximum interface height (\(W_{{\text{m}}}\)):⁴⁴

In Fig. 17, at the p-Si/CoPc interface, the \(m\) values (negative) decline as temperature increases, while the values of \(W_{{\text{m}}}\) increase as temperature increases. Factors such as the applied voltage, doping concentration, and temperature have an influence on the values of \(W_{{\text{m}}}\), which are equivalent to the difference between the work functions of each semiconductor material, p-Si and CoPc.⁴⁴

At room temperature, \(W_{{\text{m}}}\) is 0.11 ± 0.01 eV, while at medium frequencies, the values of \(W_{{\text{m}}}\) and \(m\) are 0.15 ± 0.01 eV and 0.71 ± 0.01 for room temperature and all temperatures, respectively. At the Al/p-Si interface, \(W_{{\text{m}}}\) can be considered as equivalent to subtracting the work function of the metal \(\varphi_{{{\text{Al}}}}\) from the electron affinity of the semiconductor \(\chi_{{{\text{Si}}}}\). To construct a semiconductor junction with various band gaps (E_g), work functions (Φ), and electron affinities \( \left( {\chi_{Si} } \right)\), we should note that the discontinuities in the energy bands are at equilibrium as the Fermi levels line up. In the ideal case, according to the Anderson affinity rule,⁴³ the difference in the electron affinity \(q(\chi_{Si} - \chi_{CoPc} )\) gives the discontinuities in the conduction band (ΔE_c), and the difference in \(q(E_{g} - E_{c} )\) gives the discontinuities in the valance band (ΔE_v).

Figure 18a, b shows the operation mechanism of the tested hybrid heterojunction with the help of energy band diagrams. The electron affinity (lowest unoccupied molecular orbital, LUMO) and ionization potential (highest occupied molecular orbital, HOMO) values used in this diagram were taken from the literature.^2,10,44 To realize the thermal equilibrium, holes concerning the wide-band-gap material (CoPc) will flow into the narrow one (p-Si), producing hole aggregation in the narrow-band-gap material at the interface (p-Si/CoPc).^10,16

For the Al/p-Si/CoPc/Au structure, the values of ΔE_c and ΔE_v were estimated as 0.69 and 0.11 eV, respectively. The voltage V_bi developed at the p-Si/CoPc interface because of the variation in work function is estimated as:¹⁶where Φ_CoPc and Φ_Si are the work functions of CoPc and silicon, respectively. At room temperature, at the p-Si/CoPc interface, the voltage V_bi is corresponding to the maximum barrier height \(W_{{\text{m}}}\). The CoPc work function Φ_CoPc can be estimated as follows:

The work function Φ_CoPc is found to be 5.09 eV. Thus, the contacts between the p-type semiconductor (CoPc) and Au electrode are ohmic metal–semiconductor contacts, where Φ_Au > Φ_CoPc.²⁸ The Schottky barrier formation at the metal–semiconductor interface (Al/p-Si) has been a subject of debate for decades. The Schottky–Mott model²⁴ suggested that the Schottky barrier height \(\Phi_{b}\) is equivalent to the variations between the work function of the metal Φ_Al and the electron affinity of the semiconductor \( \Phi_{{{\text{Si}}}}\). Therefore, the value of \(\Phi_{b}\) can be estimated from the energy band profile as 0.14 eV, which in correlation with the \({\text{W}}_{{\text{m}}}\) value at (Al/p-Si interface) moderates the frequency. In this research, we found that Φ_Si < Φ_CoPc, χ_Si > \(\chi_{CoPc}\), and \(\chi_{Si}\) + E_gSi < \(\chi_{CoPc}\) + E_gCoPc. Therefore, the band profile for the Al/p-Si/CoPc/Au heterostructure can be built¹⁶ as indicated in Fig. 18b.

More useful information about the elements with the lowest capacitance can be obtained using the complex electric modulus. In this work, the complex electric modulus was estimated to give a better explanation of the relaxation process in dielectric specimens. The complex electric modulus (\({M}^{*}\)) can be estimated from complex permittivity:^37,45where M′ is the real part and M″ is the imaginary part of the complex electric modulus. Figures 18 and 19 illustrate the changes in \(M^{\prime} \) and \(M^{\prime\prime} \) as frequency changes, respectively. M′ approaches zero at the low frequency limit; a small tail is present, perhaps owing to the large capacitance of the electrodes.^39,46

As the temperature increases, M′ dispersion increases until reaching a frequency of 740 kHz; beyond this frequency, they merge and decline. When the temperature increases, the relaxation peaks of M″ move towards higher frequencies, as shown in Fig. 19. To estimate the relaxation time in the most proper way, the inverse of the frequency of the maximum position should be taken. Figure 20 indicates the variation in the relaxation time ln (\({\tau }_{\mathrm{m}}\)) versus 1000/T.

This change is explained by the Arrhenius law, as in Eq. (6). Table II shows the values of ∇Eτ, which were estimated from the slope of the linear fit. The behavior of both Z″ and M″ at 353 K is depicted in Fig. 21. Z″_max and M″_max do not occur at the same frequency, indicating a wide distribution of the relaxation times and a non-Debye type of relaxation.

The relaxation time Z″ is associated with \({\tau }_{2}\) at the p-Si/CoPc interface, while the relaxation time M″ is related to \({\tau }_{1}\) at the Al/p-Si interface (conducting volume).^47,48 The more rapid movement of mobile carriers contributes to the DC conductivity of the tested device and is evidence of a lower relaxation time.

The dependence of total electrical conductivity \({(\sigma }_{\mathrm{tot} })\) on frequency (\(\upomega \)) can be expressed by the following relationship:⁴⁹where \({\upsigma }_{{{\text{dc}}}}\) and \(\sigma_{{{\text{ac}}}}\) are the DC and AC conductivity analogous to zero and high frequencies, respectively. Figure 22 depicts the variation in total conductivity in relation to frequency. Over the frequency range of 100 Hz to 8 kHz, the values of \(\sigma_{{{\text{dc}} }}\) become almost constant; however, they increase as the temperature increases. This decreases the binding force through exciton formation. Charge carrier mobility increases as the temperature increases. At the frequency limit, according to Coulomb’s law, the dielectric constant increases as the temperature increases, so the binding energy declines. The DC conductivity values (\(\sigma_{{{\text{dc}}}}\)) increase as frequency (\(\omega\)) increases because polarization declines as frequency increases. When AC conductivity (\(\sigma_{{{\text{ac}}}}\)) increases, this causes an increase in the eddy current, which in turn increases the energy loss.⁴⁷ It then gradually declines due to series resistance.⁵⁰ At zero frequency for various temperatures, \(\sigma_{{{\text{dc}}}}\) can be found from an extrapolation of the experimental data to the zero frequency of \(\sigma_{{{\text{tot}}}}\). The Arrhenius law can be used to show the temperature dependence of \(\sigma_{{{\text{dc}}}}\):⁵¹where \(\sigma_{0}\) is the DC conductivity’s pre-exponential coefficient and \(\nabla E_{{{\text{dc}}}}\) is the activation energy for conduction (Fig. 23).