1 Introduction

Through the last decades, there were intensive efforts for replacing the inorganic semiconductors by organics in most of the electronic applications. Organic semiconductors overcome many of inorganic disadvantages such as high cost on a large scale in either preparation or device fabrication and the low compatibility with flexible substrates [1]. The high optical absorption, high flexibility, and the ability to tune the electronic and optoelectronic properties of the material by controlling the molecular structure make these materials promising for a wide scale of potential applications [1,2,3]. Although the (opto-)electronic devices based on organic semiconductors have emerged the electronic devices markets until now, inorganic semiconductors are favorable for most of the sophisticated electronic applications [4, 5]. In this framework, one can notice a great acceleration in the field of scientific researches concerning the material science and characterization of organic semiconductors aiming to modify their properties for eventually recording a high-performance efficiency of the electronic device. Among the different categories of organic semiconductors, two-dimensional organic oligomers attract great interest in many applications due to their planar structure and conjugation that cause higher electronic delocalization [6, 7].

1,4,5,8-Naphthalenetetracarboxylic-dianhydride with a planar π-stacking structure has emerged as a promising material for the organic electronics community. Over the past years, NTCDA attracted the attention of many researchers with the aim to study the epitaxial growth of several monolayers of this molecule, organic–inorganic hybrid complexation, and degradation mechanisms [8,9,10,11,12,13,14,15,16,17,18,19]. Among several n-type materials, NTCDA is considered one of the most promising organic semiconductors due to its high mobility, high air stability, and strong photocurrent gain in optical photodetectors [20,21,22,23,24]. Moreover, according to the innate nature of high absorption of NTCDA, this is a significant property that qualifies it to be utilized as an active layer in photodiode applications [25,26,27]. In this sense, NTCDA was exploited in a wide range of electronic and optoelectronic applications including organic solar cells [28, 29]. The NTCDA thin film of thickness (30–50) nm have played rules as an optical spacer and electron transport layer (ETL) in organic solar cell of the following architecture [ITO/novaled p-dopant 2/N,N,N,N-tetrakis(4-methoxyphenyl)-benzidine/zinc-phthalocyanine-doped fullerene (1:1)/fullerene/NTCDA:acridine orange base (20:1)/AL] showed a clear enhancement in the cell efficiency reaching the maximum value 2.83% at thickness 30 nm [28]. Furthermore, NTCDA was exploited with 3,4,9,10 perylenetetracarboxylic bisbenzimidazole (PTCBI) due to the equality of LUMO position of both (~ 4 eV) for modifying the organic solar cell based on 2,4-bis[4-(N-phenyl-1-naphthylamino)-2,6-dihydroxyphenyl] squaraine/fluorine and showed efficiency enhancement by > 25% relative to the conventional architectures that use bathocuproine [29]. NTCDA is considered as one of the common n-type materials that are used in n-channel organic field-effect transistors (OFETs) [20, 30,31,32] and recorded a charge mobility ~ 0.016 cm2 V−1 s−1, voltage threshold ~ 32 V, cut-off current ~ 1.76 nA, and saturation on/off ratio ~ 2.25 × 102 at (drain–source voltage = 60 V) [30]. Organic diodes based on NTCDA/polypyrrole and NTCDA/poly(3,4-ethylenedioxythiophene) doped with poly(styrenesulfonate) showed a rectification behavior of rectification ratio ~ 4.1 × 103 and break down at 9 V with turn-on voltage ~ 1.2 and 1.7 V, respectively [33]. After the first deep investigation of photocurrent multiplication in NTCDA [34], NTCDA becomes a favorable applicable material for photodetection devices. Therefore, much previous work concerned the exploitation of NTCDA with p-type organic semiconductor for optoelectronic applications, where NTCDA/PTCDA multiple quantum well-organic photodetector [35] and NTCDA/phthalocyanine heterojunction light sensor [36] achieved a good performance. Interestingly, NTCDA showed a decisive role in enhancing the performance of organic light-emitting diodes (OLEDs) based on ITO/NPB/Alq3 + C54T/Alq3/LiF/Al, when inserted between ITO and NPB for enhancing the process of hole injection due to the strong interaction between NTCDA and ITO that lowers the barrier for holes that transfer from ITO to hole transporting layer (NPB) [37]. The process of insertion NTCDA thin layer (3 nm) between ITO and NPB achieved a decrement of operating voltage of OLED from 12.2 to 9.2 V (J ~ 100 mA cm−2) which gives rise to 104 cd cm−2 luminance.

Despite the growing interest in the physical properties of this compound and how to exploit it in a lot of architectures as an effective n-type material to achieve high efficiency, there is no previous work concerned exploring the nature of inter layering NTCDA in metal/semiconductor, MS, and Schottky diode. Hence, this research focuses on the role of NTCDA thin film on the performance of the Al/p-Si Schottky diode. According to the high thermal stability of NTCDA, as obtained in our previous work [26], the performance of the fabricated Schottky diode is investigated in the temperature range (303–383) K.

2 Experimental work

2.1 Schottky diode fabrication procedures

The architecture of the fabricated Schottky diode is shown in Fig. 1a. 1,4,5,8-Naphthalenetetracarboxylic-dianhydride, NTCDA, (C14H4O6) powder purchased from Alfa Aeser Company of chemical structure that is shown in Fig. 1b is our raw material. A thin film of NTCDA of thickness 150 nm was deposited on chemically pre-etched p-type silicon substrates of orientation [111] and resistivity (8–13 Ω cm) using thermal evaporation technique (coating unit Edwards 306A-England) at room temperature under pressure 3 × 10–5 mbar. The deposition rate and film thickness were controlled using the quartz crystal monitor thickness model Edward FTM6 to be 0.2 nm s−1 and 150 nm, respectively. Then, a 100 nm aluminum top electrode was deposited on NTCDA thin film using a shadow mask in the form of a mesh. Finally, a thick-aluminum bottom electrode of thickness 300 nm was deposited on the backside of Si.

Fig.1
figure 1

a Architecture of the fabricated Schottky diode, b molecular structure of NTCDA, and c, d 3D AFM image of as-deposited NTCDA thin film

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2.2 Characterization techniques

Atomic Force Microscope, AFM, operated in contact mode using a nonconductive silicon nitride probe, manufactured by Bruker (Model: MLCT-MT-A) was employed to investigate the surface topography of NTCDA as-deposited thin film. The scan area under study was 5 × 5 μm2 with 1 Hz scan rate. The microstructural properties of the prepared thin film were investigated using a high-resolution transmission electron microscope, TEM; model JEOL-JEM-HR-2100 with accelerating voltage 200 kV. The obtained TEM images were in DM3 format and were analyzed using ImageJ software. The electrical properties of the fabricated Al/NTCDA/p-Si/Al diode were investigated by measuring the I–V characteristic curve under the dark conditions from − 2.5 to + 2.5 V using Keithley electrometer model 6517B at different temperatures ranges from 313 to 383 K.

3 Results and discussion

3.1 Topographical and microstructural properties of NTCDA thin films

The surface roughness of organic interfacial thin film is a primary factor that has a significant impact on the performance of any electronic device [38,39,40], where surface nanoroughness induces surface defects that play an important role in charge dynamics and recombination rate at surface [41]. Hence, the surface topography of NTCDA thin film was inspected using the atomic force microscope. The AFM images of NTCDA thin film that is shown in Fig. 1c, d reveal the granular nature of the NTCDA surface with a root-mean-square roughness and average roughness about 4.05 nm and 3.15 nm, respectively. The small values of roughness parameters clarify the validity and suitability of forming a smooth interfacial contact between NTCDA and the metal electrode which enhances the performance of any electronic device [42, 43].

Furthermore, the microstructural properties of the organic interlayer film between the semiconductor and metal electrode have essential importance for achieving output performance outstanding [44]. In this essence, the microstructural properties of NTCDA thin film are studied using TEM. Figure 2 reveals the TEM image of NTCDA thin film and shows a good quality crystallized structure with small molecular crystals arranged with a definite grain boundary. There is a distribution of pores with different dimensions. The average crystallite size as estimated from TEM is about 89 nm.

Fig.2
figure 2

The TEM micrograph of As-deposited NTCDA thin film

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3.2 Current–voltage characteristics

Figure 3 shows the temperature-dependent I–V characteristics of the fabricated Schottky diode. The observed electrical response shows rectification behavior with a room temperature–rectification ratio of 4.6 at ± 2 V. The forward current increases linearly at low forward biasing voltages, and then under the influence of the high series resistance, the device behavior deviates from linearity to an exponential growth behavior. It is clear that the temperature variation has a significant impact on the electrical properties of the fabricated device. The increment of the produced current with increasing the temperature at a certain constant voltage indicates the negative temperature coefficient of resistivity [45]. The cut-in voltage of the fabricated device ranged from 1.3 to 1.19 V as the temperature increases from 303 to 383 K.

Fig.3
figure 3

I–V characteristic curve of Al/NTCDA/p-Si/Al Schottky diode, while the inset represents the semi-logarithmic I–V relation at different temperatures

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The sensitivity of the fabricated Schottky diode toward the temperature variation is discussed in terms of temperature coefficient (TC) of either voltage or current [46]. The TC value can be estimated from d[lnS(T)]/dT, where S(T) can be represented as forward current, IF, or forward voltage, VF [46]. Figure 4a, b represents the temperature dependence of forward current and voltage, respectively. The estimated values of the temperature coefficient of current ranges from 1.24 to 0.92% K−1 at voltage range from 0.5 to 2.5 V as revealed in Fig. 4a, while the temperature coefficient of voltage ranges from − 0.59 to − 0.41% K−1 at current range from 0.1 to 0.5 mA as revealed in Fig. 4b.

Fig.4
figure 4

The temperature dependence of a lnIF and b lnVF for estimating TC

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Figure 5 depicts the semi-logarithmic relation between voltage and current in forward biasing state at different temperatures. Three distinct regions are clearly observed: region I at V < 0.35 V, region II at 0.35 < V V< 1.2 V, and region III at V > 1.2 V.

Fig.5
figure 5

Semi-logarithmic plot of I–V characteristics of Al/NTCDA/p-Si/Al Schottky at different temperatures

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The thermionic emission model can be utilized to fit the linear current–voltage behavior in the low biasing region (region I). According to thermionic emission theory, the temperature dependence of I–V relation of organic/inorganic heterojunction is represented by the following relation [47, 48]:

$$I = I_{{\text{s}}} \exp\left( {\frac{{q\left( {V - IR_{{\text{s}}} } \right)}}{{nk_{{\text{B}}} T}}} \right)\left[ {1 - \exp\left( {\frac{{ - q\left( {V - IR_{{\text{s}}} } \right)}}{{k_{{\text{B}}} T}}} \right)} \right]\,\,,$$
(1)

where Is is the reverse saturation current, n is the ideality factor, q is electron charge, kB is Boltzmann constant, T is the temperature, and Rs is the series resistance. The reverse saturation current can be represented as [47, 48]:

$$I_{s} = AA^{*} T^{2} \exp \left( {\frac{{ - q\Phi_{{{\text{Bo}}}} }}{{k_{{\text{B}}} T}}} \right)\,,\,$$
(2)

where A is the device area, A* is the effective Richardson constant, and ΦBo is the barrier height at zero bias voltage which can be estimated using the following relation:

$$\Phi_{{{\text{B0}}}} = \frac{{k_{{\text{B}}} T}}{q}\ln \left( {\frac{{AA^{*} T^{2} }}{{I_{0} }}} \right)\,.\,$$
(3)

The ideality factor, n, was estimated from the slope of the linear region of the I–V curve in forward biasing using the following relation:

$$n = \frac{q}{{k_{{\text{B}}} T}}\left[ {\frac{{{\text{d}}V}}{{{\text{d}}\left( {\ln I} \right)}} - IR_{{\text{s}}} } \right].$$
(4)

The voltage across the series resistance in the low-voltage region VF < 0.35 V is negligible; hence, Eq. (4) is simplified to be:

$$n = \frac{q}{{k_{{\text{B}}} T}}\left[ {\frac{{{\text{d}}V}}{{{\text{d}}\left( {\ln I} \right)}}} \right].\,$$
(5)

Figure 6a shows the estimated values of the ideality factor and barrier height as a function of temperature. The high value of the ideality factor which is greater than unity confirms the non-ideal behavior of the fabricated device. This can be ascribed to several factors such as tunneling process [48], image-force effect [49,50,51], series resistance [52], inhomogeneity of barrier height, the high probability of electron–hole recombination within the depletion region [53, 54], and the existence of interface states due to the native layer results from SiO2 [53, 54]. These high values of ideality factor were previously recorded in many applicable devices such as V2O5/n-Si Schottky diode (n = 6.95 [55]), multiple quantum well LED GaInN/GaN (n = 5.5 [56]), Au/PEDOT:PSS/n-Si (n ~ 4.6 [57]), STO/p-Si (n = 10 [58]), and Cu(acac)2/n-Si (n = 5.6 [54]).

Fig.6
figure 6

a The temperature dependence of ideality factor and barrier height and b the conventional Richardson plot of the fabricated Schottky diode

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It was observed that the value of n decreases, while the value of ΦB increases as the temperature increases which confirms the heterogeneity of barrier height. The barrier height fluctuations is a recurrent behavior which exists in all organic/inorganic heterojunction diodes [54, 59]. The decrement of the ideality factor as a result of temperature increase ensures that charge transport across the metal/semiconductor interface is temperature-dependent. Hence, the charge carrier transportation takes place at low temperatures through surmounting only the lower energy barriers by electrons leading to a high ideality factor, while at high temperatures; electrons gained sufficient energy enough to overcome the high-energy barriers [49, 60, 61]. The insertion of the NTCDA interlayer showed a strong modification of Al/p-Si contact, where the estimated room-temperature barrier height is about 0.72 eV, which is greater than that of Al/p-Si by 0.14 eV [60].

The relation between 1000/T and ln (IS/T2) that represents the conventional Richardson plot is represented in Fig. 6b and shows a linear relation that confirms the validity of the Thermionic emission model to interpret the charge carriers conduction process of the low-voltage region.

Since the value of Schottky barrier height (SBH) is a clear manifestation of the uniformity of barrier, the temperature dependence of n and ΦB values is investigated. The clear increment of barrier height that is shown in Fig. 6a with temperature may be attributed to the variation of the semiconductor energy gap [62]. However, according to the negative temperature coefficient of the silicon energy gap, the increase of temperature will result in a narrower energy gap and hence lower barrier height which is contrary to the current situation [49, 62]. Therefore, the ΦB–T interrelation is not ascribed to the variation of the energy gap. The non-homogeneity of SBH was interpreted based on the assumption of a Gaussian distribution of SBH, P(ΦB), introduced by Werner and Güttler according to the following equations [49, 62]:

$$\Phi_{{{\text{B0}}}} = \overline{\Phi }_{{{\text{B0}}}} \left( {T = 0} \right) - \frac{{q\sigma_{{{\text{S0}}}}^{2} }}{{2k_{{\text{B}}} T}},\,$$
(6)
$$n^{ - 1} - 1 = - \rho_{2} + \frac{{q\rho_{3} }}{{2k_{{\text{B}}} T}},\,$$
(7)

where \(\overline{\Phi }_{B0}\) and \(\sigma_{s0}\) are zero-biased mean barrier height and its standard deviation, respectively, while ρ2 and ρ3 are constants that represent the potential deformation of distribution of SBH. Employing Eqs. (6) and (7), and based on the linear fitting of the relation between both ΦB0, n−1–1 and q/2kBT that revealed in Fig. 7a, the values of \(\overline{\Phi }_{B0}\),\(\sigma_{s0}\), ρ2, and ρ3 are determined and found to be 1.5 eV, 20 mV, − 0.688, and − 7.44 mV, respectively. For more confirmation, combining Eqs. (3) and (6), the conventional Richardson plot would be modified and represented as follows:

$$\ln \left( {\frac{{I_{s} }}{{T^{2} }}} \right) - \left( {\frac{{q^{2} \sigma_{{{\text{s0}}}}^{2} }}{{2k_{{\text{B}}}^{2} T^{2} }}} \right) = \ln \left( {AA^{**} } \right) - \frac{{q\overline{\Phi }_{{{\text{B0}}}} }}{{k_{{\text{B}}} T}},\,$$
(8)
Fig.7
figure 7

The relation between a n−1–1 and ΦB0 versus q/2kBT and b ln(Is/T2) − (q2σs02/2kB2T2) versus q/kBT for the fabricated Schottky diode

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where A** is the modified Richardson constant. From the linear fitting to Eq. (8) as shown in Fig. 7b, the mean barrier height and modified A** values are estimated and found to be 1.51 eV and 35.2 A cm−2 K−2. The estimated value of the mean value of barrier height (1.51 eV) is very close that estimated from Eq. (6) (1.5 eV), while the estimated modified Richardson constant (35.2 A cm−2 K−2) is in a good agreement to the known value of Richardson constant of p-Si (32 A cm−2 K−2). These coincidences confirm the validity of the thermionic emission theory and Gaussian distribution of SBHs to explain the behavior of the microelectronic parameters of Al/NTCDA/p-Si/Al Schottky diode.

The estimated high values of the ideality factor indicate that thermionic emission is no longer suitable for interpreting the transport mechanism in other voltage regions. Hence, the double logarithmic relation of forward current and voltage in region II was plotted, as shown in Fig. 8a. A set of straight lines was obtained with a slope ~ 2, indicating that the dominant conduction mechanism is space charge limited current (SCLC) controlled by a single trap state [53, 63]. The process of space charges injection from p-Si to NTCDA thin film takes place at the high forward biasing region, where the thin NTCDA film ultimately limits the dynamics of the charge due to low mobility. The space charge limited current can be expressed using the Mott–Gurney equation [64]:

$$I_{{{\text{SC}}}} = \frac{9}{8}\varepsilon \mu A\frac{{V^{2} }}{{d^{3} }}\frac{{N_{{\text{v}}} }}{{N_{{\text{t}}} }}\exp \left( {\frac{{ - E_{{\text{t}}} }}{{k_{{\text{B}}} T}}} \right),\,$$
(9)
Fig.8
figure 8

a The variation of ln(IF) versus ln(V) at different temperatures and b the temperature dependence of ln(IF) at different voltages for Al/NTCDA/p-Si/Al diode in region II

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where ε is the permittivity of NTCDA film, d is the film thickness, μ is the charge carrier mobility within NTCDA film, Nv is the effective density of states located at the edge of valance band, and Nt is the total trap density at energy level Et which locates above the valance band edge. The value of Et is determined from the slope of 1/T versus lnI at different forward voltages that are depicted in Fig. 8b. The calculated average value of Et was about 0.13 ± 0.006 eV. The estimated value of Et is smaller than that was estimated for Ph-HPQ/p-Si (0.256 eV [53]), Ch-HPQ/p-Si (0.41 eV [53]), and pyronine G(Y)/p-GaAs (0.25 eV [65]), while it is close to the values of FeTPPCl/p-Si (0.1 eV [66]) and InPcCl-D/p-Si (0.11 eV [67]).

Furthermore, at higher voltage (region III), the slope, m, of the linear relation between ln(IF) and ln(V) of the fabricated Schottky diode shown in Fig. 9a is more than 2, which suggests the space charge limited current (SCLC) dominant by an exponential trap distribution to interpret the conduction mechanism in this region [68]. In this essence, the current can be represented as follows [68,69,70,71]:

$$I = Aq\mu N_{v} \left( {\frac{{\varepsilon \varepsilon_{{\text{o}}} }}{{eP_{{\text{o}}} k_{{\text{B}}} T_{t} }}} \right)^{l} \left( {\frac{{V^{l + 1} }}{{d^{2l + 1} }}} \right)\,,\,$$
(10)
Fig.9
figure 9

a The variation of ln(IF) versus ln(V) at different temperatures and b the temperature dependence of ln(IF) at different voltages for Al/NTCDA/p-Si/Al diode in region III

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where Po represents the trap density per unit energy range at the valence band edge, ε is the dielectric constant of the material (ε = 2.89 [72]), l is a parameter related to the slope, m, by l = m − 1 and represents the ratio between (Tt/T), and Tt is a temperature parameter describes the exponential trapping distribution P(E). The values of l and Tt were calculated at different temperatures and listed in Table 1.

Table 1 The estimated diode parameters at different temperatures
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The exponential trap distribution function can be represented as follows [68, 71]:

$$P(E) = P_{{\text{o}}} \exp \left( {\frac{ - E}{{k_{{\text{B}}} T_{{\text{t}}} }}} \right),$$
(11)

where P(E) is trap density per unit energy at energy E over the edge of valance band. Hence, the total trap density, Nt, can be estimated by integrating Eq. (11) and is given by [68, 71]:

$$N_{{\text{t(e)}}} = P_{{\text{o}}} k_{{\text{B}}} T_{{\text{t}}} .\,$$
(12)

The values of both Po and Nt were determined at room temperature and different voltages using the obtained slope values of the linear relation between 1/T and ln(IF) that is shown in Fig. 9b:

$${\text{Slope}} = \frac{\Delta \ln I}{{\Delta (1/T)}} = T_{{\text{t}}} \ln \left( {\frac{\varepsilon \,V}{{qd^{2} N_{{\text{t(e)}}} }}} \right)\,.\,$$
(13)

The estimated values of Po were about 2.89 × 1044 J−1 m−3, 2.99 × 1044 J−1 m−3, and 2.97 × 1044 J−1 m−3, while Nt values were about 1.88 × 1024 m−3, 1.94 × 1024 m−3, and 1.92 × 1024 m−3 at 1.3, 1.5, and 2.5 V, respectively. These values are smaller than those that were estimated for TiOPc/p-Si [66]. The globally distributed trap density lowers the anode injection barrier, where the trapped states which are filled with injected carriers from anode enhance the screening effect that gives rise to band bending and reducing the potential difference between anode and cathode neglecting the hypotheses of the linear I–V relation [73].

The reverse current shows a strong temperature dependence, as shown in Fig. 3, which emphasizes that the leakage current is controlled by the generation–recombination of thermally activated charge carriers with Si substrate [68, 74, 75]. The dependence of reverse current, IR, on thermal activation energy, ΔE, of the charge carriers is given by [68, 74]:

$$I_{{\text{R}}} = I_{{{\text{0R}}}} \exp \left( {\frac{ - \Delta E}{{k_{{\text{B}}} T}}} \right)\,.\,$$
(14)

The estimated values of activation energy from the linear fitting of 1/T and ln(IR) relation that is depicted in Fig. 10 were about 0.199 eV which indicates that the conduction transport mechanism is tunneling of charge carriers [76].

Fig.10
figure 10

The plot ln(IR) versus1/T of Al/NTCDA/p-Si/Al Schottky diode at different reverse bias voltages

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The conduction mechanism within the Al/NTCDA/p-Si/Al Schottky diode during reverse biasing can be ascribed in terms of either Schottky emission or Poole–Frenkel emission model [45, 77]. Figure 11 depicts the relation between VR1/2and ln (IR) of the fabricated device at different temperatures and shows two linear regions of different slopes. According to Schottky conduction mechanism, the reverse current can be represented as follows [45, 77]:

$$I_{{\text{Sc,R}}} = AA^{*} T^{2} \exp \left( {\frac{{ - \Phi_{{\text{B}}} }}{{k_{{\text{B}}} T}}} \right)\exp \left( {\frac{{\beta_{{{\text{Sc}}}} E^{1/2} }}{{k_{{\text{B}}} T}}} \right),$$
(15)
Fig.11
figure 11

ln(IR) versus VR1/2for Al/NTCDA/p-Si/Al Schottky diode at different temperatures

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where E is the applied electric field and βSc is the Schottky field lowering coefficient. On the basis of the Poole–Frenkel emission mechanism, the reverse current can be represented as follows [45, 77]:

$$I_{{\text{PF,R}}} = I_{{\text{S}}} A\exp \left( {\frac{{\beta_{{{\text{PF}}}} E^{1/2} }}{{k_{{\text{B}}} T}}} \right),\,$$
(16)

where βPF is the Poole–Frenkel coefficient. The value of β is given by [44, 77]:

$$\beta = \left( {\frac{{q^{3} }}{{b\pi \varepsilon \varepsilon_{o} }}} \right)^{1/2} ,\,$$
(17)

where b is a constant equals 1 for Poole–Frenkel mechanism and equals 4 for Schottky mechanism [45, 77], ε is the dielectric constant of the material (ε = 2.89 [72]), εo is permittivity of free space (8.85 × 10–12 F m−2), and q is electron's charge. The calculated values of β were about 4.55 × 10–5 eV m1/2 V1/2 for the Poole–Frenkel mechanism and 2.28 × 10–5 eV m1/2 V1/2 for Schottky mechanism. The experimental values of β estimated from the slope of the straight lines in the region (I) and region (II) at room temperature were about 4.12 × 10–5 eV m1/2 V1/2 and 2.07 × 10–5 eV m1/2 V1/2, respectively. The difference between the calculated and experimental values of β may be due to either phonon-assisted tunneling through the barrier [75] or the small thickness of the depletion region of Schottky in NTCDA film [69]. Hence, the transport mechanism in the low-voltage region (I) is interpreted as a Poole–Frenkel mechanism, while in the high-voltage region (II) is interpreted as a Schottky emission mechanism. Both Schottky and Poole–Frenkel's lowering coefficient are temperature-dependent parameters as observed from Fig. 12. This confirms that the phonon-assisted tunneling process is the main reason for the disagreement between the theoretical value and experimental values of β.

Fig.12
figure 12

The temperature dependence of both Schottky and Poole–Frenkel lowering coefficients

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Furthermore, the series resistance of Schottky diode that arises from the contacts resistance and undepleted semiconductor resistance was calculated using modified Norde's function as follows [54, 78,79,80]:

$$F(V) = \frac{V}{\gamma } - \frac{{k_{{\text{B}}} T}}{q}\ln \left( {\frac{I(\,V)}{{AA^{*} T^{2} }}} \right),$$
(18)

where γ is the first integer (dimensionless) number greater than the ideality factor estimated from thermionic emission calculations [54, 79, 80]. From the minimum value of Norde's function F(Vo, Io), the barrier height and series resistance were calculated using the following relations:

$$R_{{\text{s}}} = \frac{{(\gamma - n)k_{{\text{B}}} T}}{{qI_{{\text{o}}} }},$$
(19)

where Io is the current value corresponds to the minimum value of Norde's function.

$$\Phi_{{\text{B}}} = qF(V_{{\text{o}}} ,I_{{\text{o}}} ) + \frac{{qV_{{\text{o}}} }}{\gamma } - k_{{\text{B}}} T.\,$$
(20)

Figure 13 shows the relation between the forward biasing voltage and Norde's function at different temperatures. The estimated values of series resistance and barrier height using Eqs. (19) and (20) are shown in Table1. A clear decrement in the series resistance values with increasing the temperature is observed.

Fig.13
figure 13

The voltage dependence of Norde's function of the fabricated Schottky diode at different temperatures

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The presence of structural and/or surface defects and organic interfacial layer between the inorganic semiconductor substrate and metal electrode produces a high density of interface states. These states play a crucial role not only in Fermi-level pinning at metal/semiconductor contact but also in the alternation of SBH values and resulting in flow of leakage current. Furthermore, the interaction between metal electrode evanescent wave function and those of organic thin-film would induce a new contribution of interface states known as the induced density of interface states, IDIS. According to Herbert Kroemer's statement in his Nobel lecture "often, it may be said that the interface is the device", and according to the essential function that engineering of interface states obtain in order to control the electrical performance of the Schottky barrier diode [61, 81,82,83,84], it was mandatorily task to give an overview of the pivotal role of NTCDA thin film on the passivation of the surface interface states and investigating their density distribution with respect to their energy. For this reason, the density of interface states, Nss, as a function of its energy, Ess, is calculated.

The voltage dependence of effective barrier height, Φe, is interrelated to the ideality factor as a function of voltage by the following relation [61, 81,82,83,84]:

$$\frac{{{\text{d}}\Phi_{{\text{e}}} }}{{{\text{d}}V}} = 1 - \frac{1}{n\left( V \right)},\,$$
(21)

In case of insertion of a sufficiently thick organic layer, the ideality factor and interface states are related through the following relation [61, 81,82,83,84]:

$$\frac{1}{n(V)} = \frac{{\varepsilon_{i} }}{{\varepsilon_{i} + q^{2} N_{{{\text{ss}}}} \delta }},\,$$
(22)

where ε is the dielectric constant of the NTCDA layer and δ is its thickness. The effective barrier height is related to the estimated barrier height ΦB by the following equation [61, 81,82,83,84]:

$$\phi_{{\text{e}}} = \phi_{{\text{B}}} + \left( {1 - \frac{1}{n(V)}} \right)V.$$
(23)

The interface states energy, Ess, with respect to the topmost of p-type semiconductor valance band is expressed as follows [61, 81,82,83,84]:

$$E_{{{\text{ss}}}} - E_{{\text{v}}} = q\left( {\phi_{{\text{e}}} - V} \right).\,$$
(24)

Figure 14 reveals the exponential growth of the density of interface states in energy ranges from the middle of the gap to the top of the valance band for the fabricated device at different temperatures. It is noted that the density of states ranges from 4.3 × 1013 to 1.5 × 1014 eV−1 cm−2 and from 3.5 × 1013 to 1.3 × 1014 eV−1 cm−2 for 303 K and 383 K, respectively. The resulted values of Nss are comparable to the obtained values in the previous work [83, 84] and higher than that was obtained by ref [85].

Fig.14
figure 14

The distribution of interface states for Al/NTCDA/p-Si/Al Schottky diode at different temperatures and the inset shows the purposed interaction of NTCDA with Si surface dangling bonds

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It can be noticed from Fig. 14 that the distribution of interface states density that exists in charge-neutral level, CNL, above Fermi level as shown in Fig. 15 at a certain temperature decreases as their energy position shifts away from the top of valance band till it reaches an optimum value where the density of interface states reach a minimum value and then begin to increase again. Moreover, the temperature shows a significant impact on the passivation of interface states, whereas the temperature increases, the density of interface states decreases slightly and the optimum value energy position shifts upward away from the top of the valance band. The obtained temperature dependence of the interface state density was previously observed at Er/p-InP Schottky diode [86] and V(200 Å)/Pd(300 Å)/InP Schottky diode [87]. This passivation process is useful for decreasing the effect of interface states on trapping the excited charge carriers, reducing the minority charge carriers' lifetime and deteriorating the tunneling routes that were added by these states. Moreover, the passivation process would control the role of interface states in quenching the electron–hole pairs and hence increasing the photocurrent of any optoelectronic device. The passivation of interface states density of AL/NTCDA/p-Si under the influence of temperature is convenient to the results of the series resistance decrease of the fabricated Schottky diode with increasing temperature.

Fig.15
figure 15

The purposed energy-band diagram of the fabricated Schottky diode under thermal equilibrium condition

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The offset between the conduction band of p-Si and LUMO energy limits the migration of electron from the conduction band of p-Si to LUMO of NTCDA, hence decreasing the recombination at the electrode. The passivation process involves the passivation of the dangling bonds on the Si substrate's surface. It may be proposed that after depositing NTCDA thin film, the dangling bonds of Si surface would be interacted with oxygen atom of the acyl groups in NTCDA due to the high chemical affinity of Si to oxygen through a covalent bond resulting in partial passivation as depicted in the inset of Fig. 14. This passivation would decrease the density of interface states which act as electron–hole recombination sites [88]. Hence, NTCDA acted as a partial passivation layer in addition to conserving the conjugation characteristics.

Consider the case, where organic and inorganic semiconductors that are different in work function, energy gap, and electron affinity are brought together forming an organic/inorganic heterojunction, the valance band of inorganic and HOMO of organic will offset by a discontinuity, ΔEV, while the conduction band of inorganic and LUMO of organic will offset by, ΔEC [53]. These discontinuities depend on the difference in the energy gap of both, ΔEg = Eg(Si) − Eg(NTCDA), and energy that the electron should have to leave the conduction or LUMO of inorganic or organic, respectively, to vacuum level, the electron affinity, χ. Figure 15. reveals the purposed energy diagram of the fabricated Schottky diode. The non-applicability of the concept of charge flow from the low to the high work function side in organic/inorganic interfaces means leads to the alignment of the vacuum level (Δ = 0) according to Schottky–Mott rule or Anderson rule [53, 89].

According to Anderson rule [53, 89] and based on the known values of silicon energy gap (Eg Si = 1.12 eV), electron affinity (χSi = 4.05 eV), and work function (ΦSi = 4.97 eV) [90] and with the help of the previously estimated values of NTCDA frontiers molecular orbitals parameters such as electron affinity (χNTCDA = 4 eV) [91] and energy gap (Eg NTCDA = 3.3 eV) [92], the discontinuities ΔEV and ΔEC are estimated, where (ΔEC = χSi − χNTCDA) and (ΔEV = ΔEg − ΔEC)[53] and expressed on Fig. 15. Moreover, the work function of NTCDA is estimated from the built-in potential, Vbi, barrier height, ΦB, and, Vp, which represents the difference in potential between Fermi level and the valance band top by the following equation [53]:

$$V_{{{\text{bi}}}} = \Phi_{{{\text{Si}}}} - \Phi_{{{\text{NTCDA}}}} = \Phi_{{\text{B}}} - V_{{\text{P}}} ,$$
(25)

Taking into account that the value of Vp was estimated to be 0.228 eV [53] and ΦB equals 0.72 eV, the work function of NTCDA is calculated and found to 4.48 eV. Since the work function and electron affinity of silicon are greater than these of NTCDA, while NTCDA has a wider energy gap than that of silicon, and hence, the fabricated device behaves as an accumulating heterojunction. Under thermal equilibrium and zero-biasing, the major charge carriers of NTCDA will be transported by Thermionic emission through the barrier between the heterojunction, while the minor charge carriers would tunnel through the barrier. Since the valance band offset is larger than conduction band offset, hence the current flow through the fabricated device will be predominantly determined by the electron flow from NTCDA to p-Si [93]. The obtained behavior of I–V curve depicted in Fig. 3 is a result of charge carriers' diffusion in CNL region beside the charges recombination at the space charge region at the interface. The interface states that exist in the CNL region will play a significant role in the recombination probability between the generated electron–hole pairs [89].

4 Conclusion

The performance of MIS Schottky diode was investigated with the existence of 1,4,5,8 naphthalenetetracarboxylic dianhydride, NTCDA, thermally evaporated thin film as an interlayer between the top Al electrode and the p-Si substrate. The structural and topographical properties of the fabricated film were investigated and showed the nanostructured nature of the film and the low values of surface roughness. Furthermore, both barrier height, ideality factor, and the series resistance of the present architecture were calculated at different temperatures. The charge carriers' conduction mechanism within the fabricated device is found to be SCLC conduction in forward biasing. The density of interface states is estimated and found to be in the range (4.3 × 1013–1.5 × 1014) eV−1 cm−2 and (3.5 × 1013–1.3 × 1014) eV−1 cm−2 for 303 K and 383 K, respectively. The energy shift of Nss distribution under the influence of temperature was interpreted in terms of surface passivation of Si surface defects. This confirms that the peripheral temperature tunes the performance of the present architecture which suggests this device as a temperature sensor. The sensitivity of the fabricated device as a temperature sensor was evaluated in terms of the temperature coefficient of current and found to range from 1.24 to 0.92% K−1 at voltage range from 0.5 to 2.5 V, while the temperature coefficient of voltage was ranged from − 0.59 to − 0.41% K−1 at current range from 0.1 to 0.5 mA.