# Analysis of barrier inhomogeneities in Ti/p-type strained Si0.95Ge0.05 Schottky diodes using reverse current-voltage characteristics

Mohammed Mamor, Khalid Bouziane, Hind Chakir, Pierre Ruterana

► To cite this version:

Mohammed Mamor, Khalid Bouziane, Hind Chakir, Pierre Ruterana. Analysis of barrier inhomogeneities in Ti/p-type strained Si0.95Ge0.05 Schottky diodes using reverse current-voltage characteristics. Materials Science in Semiconductor Processing, 2024, 176, pp.108314. 10.1016/j.mssp.2024.108314. hal-04622015

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https://normandie-univ.hal.science/hal-04622015

Submitted on 24 Jun 2024

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# Analysis of barrier inhomogeneities in Ti/p-type strained Si0.95Ge0.05 Schottky diodes using reverse current-voltage characteristics

Mohammed Mamor, Khalid Bouziane, Hind Chakir, Pierre Ruterana

► To cite this version:

Mohammed Mamor, Khalid Bouziane, Hind Chakir, Pierre Ruterana. Analysis of barrier inhomogeneities in Ti/p-type strained Si0.95Ge0.05 Schottky diodes using reverse current-voltage characteristics. Materials Science in Semiconductor Processing, 2024, 176, pp.108314. 10.1016/j.mssp.2024.108314. hal-04622015

HAL Id: hal-04622015

https://normandie-univ.hal.science/hal-04622015v1

Submitted on 24 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Analysis of barrier inhomogeneities in Ti/p-type strained Si<sub>0.95</sub>Ge<sub>0.05</sub> Schottky diodes using reverse current-voltage characteristics

Mohammed Mamor<sup>a,c</sup>, Khalid Bouziane<sup>b</sup>, Hind Chakir<sup>a,b</sup> and Pierre Ruterana<sup>c</sup>

<sup>a</sup>Laboratoire Physique Fondamentale et Appliquée–Safi (LPFA-Safi), Faculté Poly-Disciplinaire, B. P. 4162 Safi, Université Cadi Ayyad, Marrakech, Morocco.

<sup>b</sup>LERMA, Collège Ingénierie et Architecture, Université Internationale de Rabat, Parc Technopoles Rabat-Shore, 11100 Sala El Jadida, Morocco

<sup>c</sup>CIMAP, UMR 6252, CNRS, ENSICAEN, Université de Caen-Normandie, CEA, 6 Boulevard du Maréchal Juin, 1400 Caen, France.

## ABSTRACT

Extracting electrical parameters such as barrier height inhomogeneities (BHi) from the forward bias can be very challenging in metal/semiconductor (M/Sc) contacts characterized by low barrier height and high series resistance. In this work we demonstrate that the reverse bias characteristics can be used as an efficient alternative method to investigate the inhomogeneity of low barrier height in M/Sc contacts. In particular, the BHi in Ti/p-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> Schottky barrier diodes (SBD) has been investigated using reverse current–voltage-temperature (I<sub>R</sub>–V<sub>R</sub>–T) characteristics over the temperature range of 100-300 K. The temperature dependence of the measured barrier heights (Φ<sub>RBp</sub>) using reverse-bias is successfully explained in terms of a thermionic emission (TE) current transport associated with Gaussian distribution of the barrier height due BHi. The mean homogeneous barrier height (Φ̄<sub>RBp</sub>) and its corresponding standard deviation (σ<sub>Rs</sub>) were determined for different reverse voltages. A decrease in Φ̄<sub>RBp</sub> and an increase in σ<sub>Rs</sub> with increasing reverse bias are observed, indicating a large barrier height distribution upon high applied reverse bias. The deduced average zero-field barrier height (Φ<sub>FBp</sub> ≈ 0.59 eV) from I<sub>R</sub>–V<sub>R</sub> agrees well with the corresponding value determined from the forward characteristics. Moreover, the reverse bias extracted values for Richardson constant A* based on BHi model, are found in fair agreement with the reported theoretical value of 32 A·cm<sup>-2</sup> K<sup>-2</sup> for our p-type strained Si<sub>0.95</sub>Ge<sub>0.05</sub> alloy. Finally, the results obtained on Pd/n-type Si<sub>0.90</sub>Ge<sub>0.10</sub> structure shown in this work fully support the utilization of such reverse bias method.

<footnote>*) Electronic mail: mohammedmamor@yahoo.com; m.mamor@uca.ac.ma</footnote>
# Analysis of barrier inhomogeneities in Ti/p-type strained Si<sub>0.95</sub>Ge<sub>0.05</sub> Schottky diodes using reverse current-voltage characteristics

Mohammed Mamor<sup>a,c</sup>, Khalid Bouziane<sup>b</sup>, Hind Chakir<sup>a,b</sup> and Pierre Ruterana<sup>c</sup>

<sup>a</sup>Laboratoire Physique Fondamentale et Appliquée–Safi (LPFA-Safi), Faculté Poly-Disciplinaire, B. P. 4162 Safi, Université Cadi Ayyad, Marrakech, Morocco.

<sup>b</sup>LERMA, Collège Ingénierie et Architecture, Université Internationale de Rabat, Parc Technopoles Rabat-Shore, 11100 Sala El Jadida, Morocco

<sup>c</sup>CIMAP, UMR 6252, CNRS, ENSICAEN, Université de Caen-Normandie, CEA, 6 Boulevard du Maréchal Juin, 1400 Caen, France.

## ABSTRACT

Extracting electrical parameters such as barrier height inhomogeneities (BHi) from the forward bias can be very challenging in metal/semiconductor (M/Sc) contacts characterized by low barrier height and high series resistance. In this work we demonstrate that the reverse bias characteristics can be used as an efficient alternative method to investigate the inhomogeneity of low barrier height in M/Sc contacts. In particular, the BHi in Ti/p-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> Schottky barrier diodes (SBD) has been investigated using reverse current–voltage-temperature (I<sub>R</sub>–V<sub>R</sub>–T) characteristics over the temperature range of 100-300 K. The temperature dependence of the measured barrier heights (Φ<sub>RBp</sub>) using reverse-bias is successfully explained in terms of a thermionic emission (TE) current transport associated with Gaussian distribution of the barrier height due BHi. The mean homogeneous barrier height (Φ̄<sub>RBp</sub>) and its corresponding standard deviation (σ<sub>Rs</sub>) were determined for different reverse voltages. A decrease in Φ̄<sub>RBp</sub> and an increase in σ<sub>Rs</sub> with increasing reverse bias are observed, indicating a large barrier height distribution upon high applied reverse bias. The deduced average zero-field barrier height (Φ<sub>FBp</sub> ≈ 0.59 eV) from I<sub>R</sub>–V<sub>R</sub> agrees well with the corresponding value determined from the forward characteristics. Moreover, the reverse bias extracted values for Richardson constant A* based on BHi model, are found in fair agreement with the reported theoretical value of 32 A·cm<sup>-2</sup> K<sup>-2</sup> for our p-type strained Si<sub>0.95</sub>Ge<sub>0.05</sub> alloy. Finally, the results obtained on Pd/n-type Si<sub>0.90</sub>Ge<sub>0.10</sub> structure shown in this work fully support the utilization of such reverse bias method.

<footnote>*) Electronic mail: mohammedmamor@yahoo.com; m.mamor@uca.ac.ma</footnote>
# I. INTRODUCTION

Metal/semiconductor (M/Sc) rectifier contacts namely Schottky barrier diode (SBD) have been for many years a subject of intense research because of their technological interest in electronic and optoelectronic integrated circuits [1-4]. Gaining a better understanding of the behavior of such M/Sc interface is of fundamental and technological importance for developing semiconductor-based devices. The electrical properties of M/Sc contacts on different semiconductors materials are characterized by their Schottky barrier heights (SBH). Numerous works have been devoted to the extraction of the Schottky parameters characterizing M/Sc contacts such as the Schottky barrier height ($\Phi_B$) and the ideality factor ($n$) [5]. These two latter parameters are usually determined either from forward current-voltage ($I_F-V_F$) or capacitance-voltage ($C-V$) measurements [6-15]. By contrast to $I_F-V_F$ characteristics of SBD at room temperature, the temperature dependence of the $I_F-V_F$ characteristics provides better information about the current conduction modes across the M/Sc interface and on the corresponding potential imperfections related to the technological steps during the manufacturing process of devices fabrication. It is worth mentioning that the analysis of the temperature dependent $I_F-V_F$ characteristics of the SBD based on thermionic emission (TE) theory reveals usually an abnormal decrease in $\Phi_B$ and an increase in $n$ with the decrease in temperature [12-21]. Attempts have been made to determine the laws which provide the temperature dependence of the two above mentioned parameters. For instance, Werner and Güttler [16] proposed a theory the so-called the inhomogeneous model. Such model explains the fluctuation of the potential barrier associated with a Gaussian distribution characterized by a mean value, ($\bar{\Phi}_B$) and a standard deviation of distribution, ($\sigma_s$) of the Schottky barrier. Several reports have interpreted their ballistic electron emission microscopy (BEEM) [22-26] data by using the Werner and Güttler model. More precisely, they demonstrated the presence of the barrier height inhomogeneities ($BHi$) at M/Sc contacts on the basis of the existence of a Gaussian distribution of the barrier heights. It should be stressed out that the forward characteristics $\ln(I_F)$ vs. $V_F$ is highly nonlinear due to high series resistance and/or the low
# I. INTRODUCTION

Metal/semiconductor (M/Sc) rectifier contacts namely Schottky barrier diode (SBD) have been for many years a subject of intense research because of their technological interest in electronic and optoelectronic integrated circuits [1-4]. Gaining a better understanding of the behavior of such M/Sc interface is of fundamental and technological importance for developing semiconductor-based devices. The electrical properties of M/Sc contacts on different semiconductors materials are characterized by their Schottky barrier heights (SBH). Numerous works have been devoted to the extraction of the Schottky parameters characterizing M/Sc contacts such as the Schottky barrier height ($\Phi_B$) and the ideality factor ($n$) [5]. These two latter parameters are usually determined either from forward current-voltage ($I_F-V_F$) or capacitance-voltage ($C-V$) measurements [6-15]. By contrast to $I_F-V_F$ characteristics of SBD at room temperature, the temperature dependence of the $I_F-V_F$ characteristics provides better information about the current conduction modes across the M/Sc interface and on the corresponding potential imperfections related to the technological steps during the manufacturing process of devices fabrication. It is worth mentioning that the analysis of the temperature dependent $I_F-V_F$ characteristics of the SBD based on thermionic emission (TE) theory reveals usually an abnormal decrease in $\Phi_B$ and an increase in $n$ with the decrease in temperature [12-21]. Attempts have been made to determine the laws which provide the temperature dependence of the two above mentioned parameters. For instance, Werner and Güttler [16] proposed a theory the so-called the inhomogeneous model. Such model explains the fluctuation of the potential barrier associated with a Gaussian distribution characterized by a mean value, ($\overline{\Phi_B}$) and a standard deviation of distribution, ($\sigma_s$) of the Schottky barrier. Several reports have interpreted their ballistic electron emission microscopy (BEEM) [22-26] data by using the Werner and Güttler model. More precisely, they demonstrated the presence of the barrier height inhomogeneities ($BHi$) at M/Sc contacts on the basis of the existence of a Gaussian distribution of the barrier heights. It should be stressed out that the forward characteristics $\ln(I_F)$ vs. $V_F$ is highly nonlinear due to high series resistance and/or the low
Schottky barrier height either on *p*-type or *n*-type semiconductor, making it difficult to extract the SBH $\Phi_{Bp}$. In fact, the value of $\Phi_{Bp}$ might be determined straight forward from the reverse current-voltage ($I_R-V_R$) characteristics by the measurements of the reverse saturation current at a given applied reverse voltage $V_R$, and by considering the *TE*, as the mode governing the reverse current transport.

The value of $\Phi_{Bp}$ is an effective barrier height which depends on the reverse voltage through the image force lowering ($\Delta\Phi_{im}$). It differs also from the zero bias barrier height as deduced from $I_F-V_F$ characteristics under the application of low electric field. Indeed, the reverse current in Schottky contact corresponds to the majority carriers from the metal side towards the semiconductor through the Schottky barrier height. By contrast, to PN junctions where the reverse bias corresponds to the minority carrier current.

Although numerous studies have been performed to investigate the *BHi* on different M/Sc contacts using $I_F-V_F$ characteristics, limited studies of $I_R-V_R$ characteristics are available [27-32]. To our knowledge, the investigation of *BHi* in M/Sc Schottky diodes using temperature dependent reverse electrical characteristics has not been reported in the literature, yet. Therewith $I_R-V_R$ characteristics may be used to investigate the *BHi* and to overcome the drawback related to the usual non-linearity of $\ln (I_F)$ versus $V_F$. This non-linearity is associated with a high series resistance and/or low Schottky barrier height.

In this work, Ti/*p*-strained $Si_{0.95}Ge_{0.05}$ is used as a test structure to investigate the *BHi* using reverse bias in the temperature range of 100–300 K. The junction of this structure has lower Schottky barrier height than the corresponding counterpart on *n*-type $Si_{0.95}Ge_{0.05}$ or *n*-type silicon.

We demonstrate in this work that the reverse method can be successfully used to extract electrical parameters of Ti/*p*-strained $Si_{0.95}Ge_{0.05}$. We show also that the utilization of the reverse bias method is corroborated by the results obtained on Pd/*n*-type $Si_{0.90}Ge_{0.10}$ Schottky barrier diode.

The choice of the Ti/*p*-strained $Si_{0.95}Ge_{0.05}$ structure is motivated by three main reasons for the purpose of this work: (i) the barrier height of the Ti/*p*-strained $Si_{0.95}Ge_{0.05}$ contact is low; since the *p*-type strained $Si_{0.95}Ge_{0.05}$ epilayer has a low band gap with respect to either the relaxed $Si_{0.95}Ge_{0.05}$
Schottky barrier height either on *p*-type or *n*-type semiconductor, making it difficult to extract the SBH $\Phi_{Bp}$. In fact, the value of $\Phi_{Bp}$ might be determined straight forward from the reverse current-voltage ($I_R-V_R$) characteristics by the measurements of the reverse saturation current at a given applied reverse voltage $V_R$, and by considering the *TE*, as the mode governing the reverse current transport.

The value of $\Phi_{Bp}$ is an effective barrier height which depends on the reverse voltage through the image force lowering ($\Delta\Phi_{im}$). It differs also from the zero bias barrier height as deduced from $I_F-V_F$ characteristics under the application of low electric field. Indeed, the reverse current in Schottky contact corresponds to the majority carriers from the metal side towards the semiconductor through the Schottky barrier height. By contrast, to PN junctions where the reverse bias corresponds to the minority carrier current.

Although numerous studies have been performed to investigate the *BHi* on different M/Sc contacts using $I_F-V_F$ characteristics, limited studies of $I_R-V_R$ characteristics are available [27-32]. To our knowledge, the investigation of *BHi* in M/Sc Schottky diodes using temperature dependent reverse electrical characteristics has not been reported in the literature, yet. Therewith $I_R-V_R$ characteristics may be used to investigate the *BHi* and to overcome the drawback related to the usual non-linearity of $\ln (I_F)$ versus $V_F$. This non-linearity is associated with a high series resistance and/or low Schottky barrier height.

In this work, Ti/*p*-strained $Si_{0.95}Ge_{0.05}$ is used as a test structure to investigate the *BHi* using reverse bias in the temperature range of 100–300 K. The junction of this structure has lower Schottky barrier height than the corresponding counterpart on *n*-type $Si_{0.95}Ge_{0.05}$ or *n*-type silicon.

We demonstrate in this work that the reverse method can be successfully used to extract electrical parameters of Ti/*p*-strained $Si_{0.95}Ge_{0.05}$. We show also that the utilization of the reverse bias method is corroborated by the results obtained on Pd/*n*-type $Si_{0.90}Ge_{0.10}$ Schottky barrier diode.

The choice of the Ti/*p*-strained $Si_{0.95}Ge_{0.05}$ structure is motivated by three main reasons for the purpose of this work: (i) the barrier height of the Ti/*p*-strained $Si_{0.95}Ge_{0.05}$ contact is low; since the *p*-type strained $Si_{0.95}Ge_{0.05}$ epilayer has a low band gap with respect to either the relaxed $Si_{0.95}Ge_{0.05}$
epilayer or Si epilayer, (ii) low Ge concentration (5%) to avoid relaxed film induced dislocations; which will deteriorate the electrical properties of the Schottky contacts, (iii) the choice of Ti characterized by a low work function ($\Phi_{Ti} = 4.3$ eV) [33] in order to obtain a good rectifier Schottky contacts on p-type materials. From application point of view, Schottky junctions of Ti on p-type strained Si<sub>0.95</sub>Ge<sub>0.05</sub> epilayers are of technological interest for infrared detector applications. Indeed, to extend the cutoff wavelength for the infrared detection, a lower barrier height is required in order to detect infrared radiation in the long wavelength region.

## II. EXPERIMENTAL DETAILS

The sample used for the SBD fabrication is a 383 nm thick epitaxial strained p-type Si<sub>0.95</sub>Ge<sub>0.05</sub> layer, doped to about $8 \times 10^{16}$ boron/cm<sup>3</sup>. It was grown by rapid thermal chemical vapor deposition (RTCVD) on a lightly doped ($4-6 \times 10^{16}$ cm<sup>-3</sup>) Si buffer layer, which in its turn was grown on an p<sup>+</sup>-type thick Si substrate. After chemical cleaning, circular Ti Schottky contacts of 0.77 mm diameter and 100 nm thick were deposited on the strained p-type Si<sub>0.95</sub>Ge<sub>0.05</sub> epilayer by electron beam evaporation. The ohmic contacts were formed on the p<sup>+</sup>-Si substrate using an eutectic In-Ga alloy. A scanning electron microscope (SEM) top view image of our typical Ti circular Schottky contact is shown in figure 1(a), and the schematic three dimensional structure of the diode is displayed in figure 1(b). The current-voltage measurements were performed on Ti/p-type strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD over a temperature range of 100-300 K using a Keithley 2400 source Ammeter. A Janis closed cycle refrigeration system model 220c was used as the low-temperature cooling-device. For good thermal contact, the sample was mounted between two copper plates having a base directly fixed on the cold finger in the cryostat. A more detailed description of the used set-up can be found in reference [34].

For comparison, Pd/n-type strained Si<sub>0.90</sub>Ge<sub>0.10</sub> SBD fabricated with the same area contacts was analyzed to confirm the validity of the reverse bias method. The n-type strained Si<sub>0.96</sub>Ge<sub>0.10</sub> alloy with a thickness of about 480 nm and with phosphorous doping concentration of about $8 \times 10^{16}$ cm<sup>-3</sup>, was grown by RTCVD on a lightly doped ($4-6 \times 10^{16}$ cm<sup>-3</sup>) Si buffer layer, which was grown on an n<sup>+</sup>-Si
epilayer or Si epilayer, (ii) low Ge concentration (5%) to avoid relaxed film induced dislocations; which will deteriorate the electrical properties of the Schottky contacts, (iii) the choice of Ti characterized by a low work function ($\Phi_{Ti}$ = 4.3 eV) [33] in order to obtain a good rectifier Schottky contacts on p-type materials. From application point of view, Schottky junctions of Ti on p-type strained Si$_{0.95}$Ge$_{0.05}$ epilayers are of technological interest for infrared detector applications. Indeed, to extend the cutoff wavelength for the infrared detection, a lower barrier height is required in order to detect infrared radiation in the long wavelength region.

## II. EXPERIMENTAL DETAILS

The sample used for the SBD fabrication is a 383 nm thick epitaxial strained p-type Si$_{0.95}$Ge$_{0.05}$ layer, doped to about $8 \times 10^{16}$ boron/cm$^3$. It was grown by rapid thermal chemical vapor deposition (RTCVD) on a lightly doped ($4-6 \times 10^{16}$ cm$^{-3}$) Si buffer layer, which in its turn was grown on an p$^+$-type thick Si substrate. After chemical cleaning, circular Ti Schottky contacts of 0.77 mm diameter and 100 nm thick were deposited on the strained p-type Si$_{0.95}$Ge$_{0.05}$ epilayer by electron beam evaporation. The ohmic contacts were formed on the p$^+$-Si substrate using an eutectic In-Ga alloy. A scanning electron microscope (SEM) top view image of our typical Ti circular Schottky contact is shown in figure 1(a), and the schematic three dimensional structure of the diode is displayed in figure 1(b). The current-voltage measurements were performed on Ti/p-type strained Si$_{0.95}$Ge$_{0.05}$ SBD over a temperature range of 100-300 K using a Keithley 2400 source Ammeter. A Janis closed cycle refrigeration system model 220c was used as the low-temperature cooling-device. For good thermal contact, the sample was mounted between two copper plates having a base directly fixed on the cold finger in the cryostat. A more detailed description of the used set-up can be found in reference [34].

For comparison, Pd/n-type strained Si$_{0.90}$Ge$_{0.10}$ SBD fabricated with the same area contacts was analyzed to confirm the validity of the reverse bias method. The n-type strained Si$_{0.96}$Ge$_{0.10}$ alloy with a thickness of about 480 nm and with phosphorous doping concentration of about $8 \times 10^{16}$ cm$^{-3}$, was grown by RTCVD on a lightly doped ($4-6 \times 10^{16}$ cm$^{-3}$) Si buffer layer, which was grown on an n$^+$-Si
substrate. It should be noted that the forward electrical characteristics of Pd/n-type strained Si0.90Ge0.10 SBD have been previously performed and are reported elsewhere [35].

### III. RESULTS AND DISCUSSION

The measured semi-log ln (I) vs. V plots in both forward and reverse directions of the Ti/p-strained Si0.95Ge0.05 SBD over the considered temperature range are shown in Fig. 2. The electrical characteristics of Ti/p-strained Si0.95Ge0.05 SBD were obtained by analyzing the I<sub>R</sub>-V<sub>R</sub>-T data at each measurement temperature under the assumption that the TE mode dominates the current transport in reverse bias. As can be seen, the Ti/p-Si0.95Ge0.05 SBD is always rectifying and obviously temperature dependent. As already reported elsewhere [20], these diodes are of high quality as demonstrated by their ideality factor close to unity. This implies that the current transport is driven by the TE mode. Hence, the variation of the current with temperature is simulated using the relationship within the TE model [36]:

$$I_{\text{TE}} = I_{\text{S}} \left[ \exp\left(\frac{qV - R_{\text{s}}I}{nK_{\text{B}}T}\right) - 1 \right] \quad (1)$$

with n is the ideality factor, R<sub>s</sub> the series resistance, V the applied voltage and I<sub>s</sub> the saturation current given by:

$$I_{\text{s}} = A^{*}ST^{2} \exp\left(-\frac{\Phi_{\text{Bp}}}{k_{\text{B}}T}\right) \quad (2a)$$

where A* is the effective Richardson constant, S the area of the diode, T the temperature of the junction, and Φ<sub>Bp</sub> the effective barrier height. A* is calculated by using the linear dependence on the Ge content. The saturation current I<sub>s</sub> is determined either by extrapolating the forward I-V characteristics to zero applied voltage or directly at reverse voltage V<sub>R</sub>. Φ<sub>0Bp</sub> stands for the zero bias barrier height when I<sub>s</sub> is deduced from forward characteristic, Φ<sub>RBp</sub> represents the reverse bias barrier height when I<sub>s</sub> is deduced from reverse characteristic. In this case, the reverse saturation current I<sub>s</sub> determined from the measurement of the current at an applied reverse voltage V<sub>R</sub> is described by:
substrate. It should be noted that the forward electrical characteristics of Pd/n-type strained Si0.90Ge0.10 SBD have been previously performed and are reported elsewhere [35].

### III. RESULTS AND DISCUSSION

The measured semi-log ln (I) vs. V plots in both forward and reverse directions of the Ti/p-strained Si0.95Ge0.05 SBD over the considered temperature range are shown in Fig. 2. The electrical characteristics of Ti/p-strained Si0.95Ge0.05 SBD were obtained by analyzing the I<sub>R</sub>-V<sub>R</sub>-T data at each measurement temperature under the assumption that the TE mode dominates the current transport in reverse bias. As can be seen, the Ti/p-Si0.95Ge0.05 SBD is always rectifying and obviously temperature dependent. As already reported elsewhere [20], these diodes are of high quality as demonstrated by their ideality factor close to unity. This implies that the current transport is driven by the TE mode. Hence, the variation of the current with temperature is simulated using the relationship within the TE model [36]:

$$I_{\text{TE}} = I_{\text{S}} \left[ \exp\left(\frac{qV - R_{\text{s}}I}{nK_{\text{B}}T}\right) - 1 \right] \quad (1)$$

with n is the ideality factor, R<sub>s</sub> the series resistance, V the applied voltage and I<sub>s</sub> the saturation current given by:

$$I_{\text{s}} = A^{*}ST^{2} \exp\left(-\frac{\Phi_{\text{Bp}}}{k_{\text{B}}T}\right) \quad (2a)$$

where A* is the effective Richardson constant, S the area of the diode, T the temperature of the junction, and Φ<sub>BP</sub> the effective barrier height. A* is calculated by using the linear dependence on the Ge content. The saturation current I<sub>s</sub> is determined either by extrapolating the forward I-V characteristics to zero applied voltage or directly at reverse voltage V<sub>R</sub>. Φ<sub>0Bp</sub> stands for the zero bias barrier height when I<sub>s</sub> is deduced from forward characteristic, Φ<sub>RBp</sub> represents the reverse bias barrier height when I<sub>s</sub> is deduced from reverse characteristic. In this case, the reverse saturation current I<sub>s</sub> determined from the measurement of the current at an applied reverse voltage V<sub>R</sub> is described by:
$$I_s \approx I_R(V_R) = A^*ST^2 \exp\left(-\frac{\Phi_{RBp}(V_R)}{k_B T}\right) \quad (2b)$$

It should be indicated that the effective barrier height $\Phi_{RBp}(V_R)$ depends on the reverse voltage bias and differs from the zero-field barrier $\Phi_{FBP}$ determined from C-V measurements under flat-band conditions. The zero-field barrier height $\Phi_{FBP}$ is related to zero bias and reverse bias barrier heights:

$$\Phi_{FBP} = \Phi_{Bp}^{I-V}(V) + \Delta\Phi_{im}(V) \quad (3)$$

where $\Delta\Phi_{im}(V)$ is the image force lowering given by [37]:

$$\Delta\Phi_{im}(V) = \left[ \frac{qE_m(V)}{4\pi\varepsilon_{sc}} \right]^{\frac{1}{2}} \quad (4)$$

with $E_m$ is the maximum electric field at the junction M/Sc, given by:

$$E_m(V_R) = \left[ \frac{2qN_A(V_{bi} + |V_R|)}{\varepsilon_{sc}} \right]^{\frac{1}{2}} \quad (5)$$

$N_A$, $V_{bi}$ and $\varepsilon_{sc}$ (= 11.93$\varepsilon_0$) being the p-type doping density, flat-band voltage, and permittivity of Si$_{0.95}$Ge$_{0.05}$ material, respectively.

As shown in Fig.2, the measured reverse current increases with increasing reverse bias in good agreement with TE model, where the SBH lowering due to image force effects was taken into account. In fact, the reverse bias dependence of the SBH deduced from the reverse characteristics was reported to exceed the expected lowering induced by image force effects [38]. In this scenario, the reverse current was considered to flow essentially through the patches exhibiting low SBHs localized at nonhomogeneous M/Sc contacts. The effective SBH $\Phi_{RBp}(V_R)$ is calculated for different applied reverse voltage bias at given temperature using equation (2b) and assuming $I_s = I_R$. The SBH deduced from the reverse bias $I_R-V_R$ data as a function of temperature is displayed in Fig. 3. The temperature dependence of the zero-bias effective SBH $\Phi_{0Bp}$ deduced from $I_F-V_F$ measurements is also displayed in this figure for comparison. It can be clearly observed that the barrier heights deduced from both the forward and reverse bias increase with the temperature. The value of SBH from reverse bias is however lower than the forward bias counterpart, regardless the temperature. Actually, the difference
$$I_s \approx I_R(V_R) = A^*ST^2 \exp\left(-\frac{\Phi_{RBp}(V_R)}{k_B T}\right) \qquad (2b)$$

It should be indicated that the effective barrier height $\Phi_{RBp}(V_R)$ depends on the reverse voltage bias and differs from the zero-field barrier $\Phi_{FBP}$ determined from C-V measurements under flat-band conditions. The zero-field barrier height $\Phi_{FBP}$ is related to zero bias and reverse bias barrier heights:

$$\Phi_{FBP} = \Phi_{Bp}^{I-V}(V) + \Delta\Phi_{im}(V) \qquad (3)$$

where $\Delta\Phi_{im}(V)$ is the image force lowering given by [37]:

$$\Delta\Phi_{im}(V) = \left[ \frac{qE_m(V)}{4\pi\epsilon_{sc}} \right]^{\frac{1}{2}} \qquad (4)$$

with $E_m$ is the maximum electric field at the junction M/Sc, given by:

$$E_m(V_R) = \left[ \frac{2qN_A(V_{bi} + |V_R|)}{\epsilon_{sc}} \right]^{\frac{1}{2}} \qquad (5)$$

$N_A$, $V_{bi}$ and $\epsilon_{sc}$ (= 11.93$\epsilon_0$) being the p-type doping density, flat-band voltage, and permittivity of Si$_{0.95}$Ge$_{0.05}$ material, respectively.

As shown in Fig.2, the measured reverse current increases with increasing reverse bias in good agreement with TE model, where the SBH lowering due to image force effects was taken into account. In fact, the reverse bias dependence of the SBH deduced from the reverse characteristics was reported to exceed the expected lowering induced by image force effects [38]. In this scenario, the reverse current was considered to flow essentially through the patches exhibiting low SBHs localized at nonhomogeneous M/Sc contacts. The effective SBH $\Phi_{RBp}(V_R)$ is calculated for different applied reverse voltage bias at given temperature using equation (2b) and assuming $I_s = I_R$. The SBH deduced from the reverse bias $I_R-V_R$ data as a function of temperature is displayed in Fig. 3. The temperature dependence of the zero-bias effective SBH $\Phi_{0Bp}$ deduced from $I_F-V_F$ measurements is also displayed in this figure for comparison. It can be clearly observed that the barrier heights deduced from both the forward and reverse bias increase with the temperature. The value of SBH from reverse bias is however lower than the forward bias counterpart, regardless the temperature. Actually, the difference
in the SBH values obtained from either the reverse or forward bias I-V characteristics is explained in terms of the image force lowering (see Table I). The increase of the barrier height as deduced from I<sub>F</sub>-V<sub>F</sub> characteristics upon temperature has been attributed to the presence of BHi at M/Sc contacts [6, 12-21]. A similar behavior in the reverse bias mode is observed as shown in Fig. 3. Like the forward mode, the Werner-Güttler model [16] has been used to determine the reverse bias mean value (Φ<sub>RBp</sub>) and its standard deviation (σ<sub>Rs</sub>):

$$ \Phi_{RBp}(V_R) = \bar{\Phi}_{RBp}(V_R) - \frac{q^2 \sigma_{Rs}^2(V_R)}{2k_B T} \quad (6) $$

As shown in Fig. 4, the SBH measured at different reverse bias fits linearly with the inverse of temperature, in good agreement with equation (6). This is a direct consequence of the presence of lateral BHi at Ti/p- strained Si<sub>0.95</sub>Ge<sub>0.05</sub> interface. From the straight lines of the plots displayed in Fig. 4, the reverse bias mean barrier height Φ<sub>RBp</sub> and its standard deviation σ<sub>Rs</sub> is deduced from the intercept at the ordinate and the slope, respectively. The corresponding values for different applied bias voltage are plotted in Fig. 5 and summarized in Table I. One can notice a decrease of Φ<sub>RBp</sub> and an increase of σ<sub>Rs</sub> when the applied reverse bias increases. The decrease of Φ<sub>RBp</sub> with increasing reverse bias is consistent with the respective increase of Schottky barrier lowering due to the image force effects. The image force lowering is calculated according to equation 4 as a function of the reverse bias, by taking into account the doping density in the p-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> epilayer. The calculated image force lowering ΔΦ<sub>im</sub>(V<sub>R</sub>) values at different reverse bias V<sub>R</sub> are also listed in Table I. After including the image force lowering effect correction, the obtained SBH (Φ<sub>RBp</sub> + ΔΦ<sub>im</sub>(V<sub>R</sub>)) is equivalent to the Schottky barrier height determined under flat band conditions. The corrected SBH is found to be independent of the applied reverse bias. Therefore, the measured reverse mean barrier height values are modified solely by the image force effects. This confirms that the TE mode dominates the conduction transport mechanism in the reverse current-voltage characteristics. Moreover, a correlation between the increases of σ<sub>Rs</sub> with reverse bias is well observed. This indicates that the inhomogeneities of SBH can be interpreted in terms of the variation of the barrier height
in the SBH values obtained from either the reverse or forward bias I-V characteristics is explained in terms of the image force lowering (see Table I). The increase of the barrier height as deduced from I<sub>F</sub>-V<sub>F</sub> characteristics upon temperature has been attributed to the presence of BHi at M/Sc contacts [6, 12-21]. A similar behavior in the reverse bias mode is observed as shown in Fig. 3. Like the forward mode, the Werner-Güttler model [16] has been used to determine the reverse bias mean value (Φ<sub>RBp</sub>) and its standard deviation (σ<sub>Rs</sub>):

$$ \Phi_{RBp}(V_R) = \bar{\Phi}_{RBp}(V_R) - \frac{q^2 \sigma_{Rs}^2(V_R)}{2k_B T} \qquad (6) $$

As shown in Fig. 4, the SBH measured at different reverse bias fits linearly with the inverse of temperature, in good agreement with equation (6). This is a direct consequence of the presence of lateral BHi at Ti/p- strained Si<sub>0.95</sub>Ge<sub>0.05</sub> interface. From the straight lines of the plots displayed in Fig. 4, the reverse bias mean barrier height Φ<sub>RBp</sub> and its standard deviation σ<sub>Rs</sub> is deduced from the intercept at the ordinate and the slope, respectively. The corresponding values for different applied bias voltage are plotted in Fig. 5 and summarized in Table I. One can notice a decrease of Φ<sub>RBp</sub> and an increase of σ<sub>Rs</sub> when the applied reverse bias increases. The decrease of Φ<sub>RBp</sub> with increasing reverse bias is consistent with the respective increase of Schottky barrier lowering due to the image force effects. The image force lowering is calculated according to equation 4 as a function of the reverse bias, by taking into account the doping density in the p-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> epilayer. The calculated image force lowering ΔΦ<sub>im</sub>(V<sub>R</sub>) values at different reverse bias V<sub>R</sub> are also listed in Table I. After including the image force lowering effect correction, the obtained SBH (Φ<sub>RBp</sub> + ΔΦ<sub>im</sub>(V<sub>R</sub>)) is equivalent to the Schottky barrier height determined under flat band conditions. The corrected SBH is found to be independent of the applied reverse bias. Therefore, the measured reverse mean barrier height values are modified solely by the image force effects. This confirms that the TE mode dominates the conduction transport mechanism in the reverse current-voltage characteristics. Moreover, a correlation between the increases of σ<sub>Rs</sub> with reverse bias is well observed. This indicates that the inhomogeneities of SBH can be interpreted in terms of the variation of the barrier height
distribution with applied reverse bias. Indeed, the standard deviation decreases with decreasing electric field at M/Sc interface. In fact, $\sigma_{Rs}$ decreases to zero on approaching the flat band conditions at zero-electric field.

The inhomogeneities induced barrier height fluctuations is then shown to affect considerably the temperature dependence of the reverse $I_R-V_R$ characteristics. This can be pointed out in the conventional Richardson plots, namely $\ln(I_s/T^2)$ vs. $1/k_B T$ (equation 7).

$$ \ln\left(\frac{I_R}{T^2}\right) = \ln(A^*S) - \left(\frac{\Phi_{RBp}}{k_B T}\right) \qquad (7) $$

The $\ln(I_R/T^2)$ vs. $1/k_B T$ plot is expected to show a linear behavior with a slope given by a reverse bias SBH at 0 K, $\Phi_{RBp}(T=0)$ and an intercept at the ordinate given by an experimental Richardson constant, $A^*$. From the experimental data reported in Fig.2, the value of the reverse saturation current at a given applied reverse bias $I_s$ ($V_R$) is determined for each temperature. The corresponding Richardson $\ln(I_R/T^2)$ vs. $1/k_B T$ plot for Ti/p-strained Si$_{0.95}$Ge$_{0.05}$ SBD is shown in figure 6. As can be seen, the fit of the experimental data exhibit nonlinear dependence at low temperatures ($\ln(I_R/T^2)$ axis). The values of the activation energy ($E_a = \Phi_{RBp}(T=0)$) and Richardson constant $A^*$ were deduced from the slope and the intercept at the ordinate of the linear region of these plots, respectively [39]. The results are given in Table. I. The extracted experimental values of the Richardson's constant from the conventional plots are more than two-order of magnitude lower than the theoretical value for p-type strained Si$_{0.95}$Ge$_{0.05}$ (32 A cm$^{-2}$ K$^{-2}$) [40]. The extracted SBHs are almost two-times lower than the reverse bias mean SBHs values extracted from the plot of $\Phi_{RBp}(V_R)$ vs. $1/2k_B T$ (see figure 4) at different reverse voltages. The low extracted values of $A^*$ and $\Phi_{RBp}(T=0)$ as well as the distortion in the conventional Richardson plots at lower temperature are mainly attributed to the SBH fluctuations due to inhomogeneities at the Ti/p-strained Si$_{0.95}$Ge$_{0.05}$ interface.

Several reasons have been considered to explain for the presence of BHi at M/Sc contacts. Besides interface states and/or defects that are usually at the origin of this BHi, the alloy fluctuations
distribution with applied reverse bias. Indeed, the standard deviation decreases with decreasing electric field at M/Sc interface. In fact, $\sigma_{Rs}$ decreases to zero on approaching the flat band conditions at zero-electric field.

The inhomogeneities induced barrier height fluctuations is then shown to affect considerably the temperature dependence of the reverse $I_R-V_R$ characteristics. This can be pointed out in the conventional Richardson plots, namely $\ln(I_s/T^2)$ vs. $1/k_B T$ (equation 7).

$$ \ln\left(\frac{I_R}{T^2}\right) = \ln(A^*S) - \left(\frac{\Phi_{RBp}}{k_B T}\right) \qquad (7) $$

The $\ln(I_R/T^2)$ vs. $1/k_B T$ plot is expected to show a linear behavior with a slope given by a reverse bias SBH at 0 K, $\Phi_{RBp}(T=0)$ and an intercept at the ordinate given by an experimental Richardson constant, $A^*$. From the experimental data reported in Fig.2, the value of the reverse saturation current at a given applied reverse bias $I_s$ ($V_R$) is determined for each temperature. The corresponding Richardson $\ln(I_R/T^2)$ vs. $1/k_B T$ plot for Ti/p-strained Si$_{0.95}$Ge$_{0.05}$ SBD is shown in figure 6. As can be seen, the fit of the experimental data exhibit nonlinear dependence at low temperatures ($\ln(I_R/T^2)$ axis). The values of the activation energy ($E_a = \Phi_{RBp}(T=0)$) and Richardson constant $A^*$ were deduced from the slope and the intercept at the ordinate of the linear region of these plots, respectively [39]. The results are given in Table. I. The extracted experimental values of the Richardson's constant from the conventional plots are more than two-order of magnitude lower than the theoretical value for p-type strained Si$_{0.95}$Ge$_{0.05}$ (32 A cm$^{-2}$ K$^{-2}$) [40]. The extracted SBHs are almost two-times lower than the reverse bias mean SBHs values extracted from the plot of $\Phi_{RBp}(V_R)$ vs. $1/2k_B T$ (see figure 4) at different reverse voltages. The low extracted values of $A^*$ and $\Phi_{RBp}(T=0)$ as well as the distortion in the conventional Richardson plots at lower temperature are mainly attributed to the SBH fluctuations due to inhomogeneities at the Ti/p-strained Si$_{0.95}$Ge$_{0.05}$ interface.

Several reasons have been considered to explain for the presence of BHi at M/Sc contacts. Besides interface states and/or defects that are usually at the origin of this BHi, the alloy fluctuations
in compound semiconductors may also be at the origin of BHi. It has been previously reported that BHi at Au/AlInGaN Schottky contacts with different indium and aluminum content is mainly due to alloy fluctuation [14]. In our case, the Ge atoms are randomly distributed within the Si₁-xGeₓ alloy and thus their distribution in the Si₀.₉₅Ge₀.₀₅ is assumed to be purely statistical. Such random distribution of Ge atoms leads to a fluctuation in the band gap energy (E_g). Since the major part of the band gap variation ΔE_g at the Si₁-xGeₓ/Si heterojunction is contained in the valence band (ΔE_g(x) = ΔE_V(x)) [41-44], the fluctuation in the band gap of Si₁-xGeₓ will induce a fluctuation in the valence band. Subsequently a fluctuation of the barrier height on p-type Si₁-xGeₓ is expected. A schematic energy band diagram between metal and p-type Si₁-xGeₓ showing fluctuation in the valence band is illustrated in Fig. 7.

In order to take account of the spatially inhomogeneous SBH, a modified Richardson plot is obtained by combining equations (2b) and (7) giving rise to equation (8):

$$ \ln(\frac{I_R}{T^2}) - (\frac{q^2 \sigma_{RS}^2}{2k_B^2 T^2}) = \ln(A^* S) - \frac{\bar{\Phi}_{RBp}}{k_B T} \quad (8) $$

The new plot ln((I_R/T^2) - (q^2 σ_{RS}^2 / 2k_B^2 T^2)) vs. 1/k_B T yields a straight line whose its slope is associated with the reverse bias mean SBH and its y-axis intercept as the new Richardson constant A* (see Fig 6). The new Arrhenius plots are linear over the whole temperature range for the investigated reverse voltages. The extracted Richardson constant shows a small bias dependence, increasing between -1.0 V and -2.5 V and followed by a slight decrease beyond (see Table 1). The mean value of Richardson constant extracted for different reverse bias (38.4 A cm⁻² K⁻²) is in fair agreement with the predicted theoretical value. On the other hand, the slope of the curves that represents the reverse bias mean SBH Φ̄_{RBp} were calculated at each reverse bias and the values match exactly, within the calculated experimental error, with those extracted from the plots Φ_{RBp} versus 1/T reported in figure 4 (Table I). We therefore conclude from this analysis that the temperature dependence of the reverse I-V-T
in compound semiconductors may also be at the origin of BHi. It has been previously reported that BHi at Au/AlInGaN Schottky contacts with different indium and aluminum content is mainly due to alloy fluctuation [14]. In our case, the Ge atoms are randomly distributed within the Si₁-xGeₓ alloy and thus their distribution in the Si₀.₉₅Ge₀.₀₅ is assumed to be purely statistical. Such random distribution of Ge atoms leads to a fluctuation in the band gap energy (E_g). Since the major part of the band gap variation ΔE_g at the Si₁-xGeₓ/Si heterojunction is contained in the valence band (ΔE_g(x) = ΔE_V(x)) [41-44], the fluctuation in the band gap of Si₁-xGeₓ will induce a fluctuation in the valence band. Subsequently a fluctuation of the barrier height on p-type Si₁-xGeₓ is expected. A schematic energy band diagram between metal and p-type Si₁-xGeₓ showing fluctuation in the valence band is illustrated in Fig. 7.

In order to take account of the spatially inhomogeneous SBH, a modified Richardson plot is obtained by combining equations (2b) and (7) giving rise to equation (8):

$$ \ln(\frac{I_R}{T^2}) - (\frac{q^2 \sigma_{RS}^2}{2k_B^2 T^2}) = \ln(A^* S) - \frac{\bar{\Phi}_{RBp}}{k_B T} \quad (8) $$

The new plot ln((I_R/T^2) - (q^2 σ_{RS}^2 / 2k_B^2 T^2)) vs. 1/k_B T yields a straight line whose its slope is associated with the reverse bias mean SBH and its y-axis intercept as the new Richardson constant A* (see Fig 6). The new Arrhenius plots are linear over the whole temperature range for the investigated reverse voltages. The extracted Richardson constant shows a small bias dependence, increasing between -1.0 V and -2.5 V and followed by a slight decrease beyond (see Table 1). The mean value of Richardson constant extracted for different reverse bias (38.4 A cm⁻² K⁻²) is in fair agreement with the predicted theoretical value. On the other hand, the slope of the curves that represents the reverse bias mean SBH Φ̄_{RBp} were calculated at each reverse bias and the values match exactly, within the calculated experimental error, with those extracted from the plots Φ_{RBp} versus 1/T reported in figure 4 (Table I). We therefore conclude from this analysis that the temperature dependence of the reverse I-V-T
characteristics of Ti/*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> contacts can be successfully explained based on *TE* model that accounts for the inhomogeneities of Schottky barrier height.

It is worth to mention that the reverse bias method has been applied to investigate *BHi* in another structure, namely Pd/*-type Si<sub>0.90</sub>Ge<sub>0.10</sub>*. The corresponding data of the reverse current versus voltage measured at temperatures between 100 and 300 K are shown in Fig. 8. The results are well interpreted on the basis of *TE* model associated with Gaussian distribution of barrier heights. The first observation that arises from the initial analysis of the measured reverse *I-V* between the two samples concerns the leakage current. As compared to Ti/*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub>*, the Pd/*-type Si<sub>0.90</sub>Ge<sub>0.10</sub>* SBD reveals an enhanced leakage current in the low temperature range (100-140 K) and at high reverse bias (3V). This may indicate that at temperature below 140 K, the presence of other current transport mechanisms such as recombination-generation current can be expected. This later mode is usually more pronounced in the reverse bias as compared to the forward bias counterpart. On the other side, the *TE* current mode might dominate the current transport in the temperature range 140-300 K. Similarly, using the same procedure as in the case of Ti/*-strained Si<sub>0.90</sub>Ge<sub>0.10</sub>*, and following the SBH distribution model proposed by Werner and Güttler, the barrier height deduced from reverse bias for Pd/*- Si<sub>0.90</sub>Ge<sub>0.10</sub>* is found to be temperature dependent (see Fig. 8b). In fact, the behavior in the reverse bias is found to be similar to that in the forward bias. To further examine the *BHi* at Pd/*-type strained Si<sub>0.90</sub>Ge<sub>0.10</sub>* the plot of *Φ<sub>RBn</sub>(V<sub>R</sub>)* vs. 1/2k<sub>B</sub>T should give a straight linear according to equation (6). Such a plot is shown in Fig. 8c and the results of the data fitting leads to the mean reverse bias *Φ̄<sub>RBn</sub>* and its standard deviation *σ<sub>Rs</sub>* as summarized in Table II. When taking into account of the image force lowering, usually more important in the reverse bias regime, the corrected SBH*Φ̄<sub>RBn</sub> + ΔΦ<sub>im</sub>(V<sub>R</sub>)* is also found independent of the applied reverse bias within the calculated experimental error. Based on the *BHi* model, a plot of the modified ln(I<sub>R</sub>/T<sup>2</sup>) - (q<sup>2</sup>σ<sub>Rs</sub><sup>2</sup>/(2k<sub>B</sub>T<sup>2</sup>)) vs. 1/k<sub>B</sub>T, according to equation (8) yields a straight line with the slope giving the reverse bias mean SBH *Φ̄<sub>RBn</sub>* and the intercept giving the modified Richardson constant *A*\*as shown in Fig. 8c. Actually, the plots of
characteristics of Ti/*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> contacts can be successfully explained based on *TE* model that accounts for the inhomogeneities of Schottky barrier height.

It is worth to mention that the reverse bias method has been applied to investigate *BHi* in another structure, namely Pd/*-type Si<sub>0.90</sub>Ge<sub>0.10</sub>*. The corresponding data of the reverse current versus voltage measured at temperatures between 100 and 300 K are shown in Fig. 8. The results are well interpreted on the basis of *TE* model associated with Gaussian distribution of barrier heights. The first observation that arises from the initial analysis of the measured reverse *I-V* between the two samples concerns the leakage current. As compared to Ti/*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub>*, the Pd/*-type Si<sub>0.90</sub>Ge<sub>0.10</sub>* SBD reveals an enhanced leakage current in the low temperature range (100-140 K) and at high reverse bias (3V). This may indicate that at temperature below 140 K, the presence of other current transport mechanisms such as recombination-generation current can be expected. This later mode is usually more pronounced in the reverse bias as compared to the forward bias counterpart. On the other side, the *TE* current mode might dominate the current transport in the temperature range 140-300 K. Similarly, using the same procedure as in the case of Ti/*-strained Si<sub>0.90</sub>Ge<sub>0.10</sub>*, and following the SBH distribution model proposed by Werner and Güttler, the barrier height deduced from reverse bias for Pd/*- Si<sub>0.90</sub>Ge<sub>0.10</sub>* is found to be temperature dependent (see Fig. 8b). In fact, the behavior in the reverse bias is found to be similar to that in the forward bias. To further examine the *BHi* at Pd/*-type strained Si<sub>0.90</sub>Ge<sub>0.10</sub>* the plot of *Φ<sub>RBn</sub>(V<sub>R</sub>)* vs. 1/2k<sub>B</sub>T should give a straight linear according to equation (6). Such a plot is shown in Fig. 8c and the results of the data fitting leads to the mean reverse bias *Φ̄<sub>RBn</sub>* and its standard deviation *σ<sub>Rs</sub>* as summarized in Table II. When taking into account of the image force lowering, usually more important in the reverse bias regime, the corrected SBH*Φ̄<sub>RBn</sub> + ΔΦ<sub>im</sub>(V<sub>R</sub>)* is also found independent of the applied reverse bias within the calculated experimental error. Based on the *BHi* model, a plot of the modified ln(I<sub>R</sub>/T<sup>2</sup>) - (q<sup>2</sup>σ<sub>Rs</sub><sup>2</sup>/(2k<sub>B</sub>T<sup>2</sup>)) vs. 1/k<sub>B</sub>T, according to equation (8) yields a straight line with the slope giving the reverse bias mean SBH *Φ̄<sub>RBn</sub>* and the intercept giving the modified Richardson constant *A*\*as shown in Fig. 8c. Actually, the plots of
experimental data shown in Fig. 8d were all well fitted using rather two straight lines instead of a
single straight line with a transition occurring at 140 K. Here again the values of *A*\* obtained from
these plots are in good agreement with the theoretical value of 112 A.K⁻².cm⁻² obtained for *n*-type
Si<sub>0.90</sub>Ge<sub>0.15</sub> [36, 37]. The temperature dependent reverse current-voltage characteristics of Pd/*n*--
strained type Si<sub>0.90</sub>Ge<sub>0.10</sub> structure is successfully explained on the basis of *TE* mechanism with a
Gaussian distribution of the barrier height. It is worth noting that the reverse mean values of barrier
height, deduced from Fig. 8d matches exactly with those obtained from the Φ<sub>RBn</sub> vs. 1/2k<sub>B</sub>T reported
in Fig 8c. We conclude that the *BHi* at Pd/*n*-type Si<sub>0.90</sub>Ge<sub>0.10</sub> using reverse electrical characteristics
can be satisfactorily explained by inhomogeneous model with Gaussian distribution of SBH.

IV. Conclusion

In this work, we have demonstrated that the temperature dependent reverse current-voltage characteristics can be used to investigate the barrier height inhomogeneities at Ti/*p*-type strained Si<sub>1-x</sub>Ge<sub>x</sub>Schottky contacts. The transport mode in the reverse bias is governed by the *TE* mode associated with a Gaussian SBH distribution. The abnormally temperature dependent of SBH deduced from reverse current-voltage characteristics is found to be similar to that using forward current-voltage characteristics. The decrease of homogeneous SBH Φ&#773;<sub>RBp</sub> and the increase of standard deviation σ<sub>Rs</sub> with increasing applied reverse bias are shown to be related to the increase of Schottky barrier lowering due to the image force effects and the high degree of inhomogeneity at Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> interface, respectively. The reverse bias values for Richardson constant *A*\*, extracted from the modified plots based on *BHi* model, are close to the theoretical value of 32 A cm<sup>-2</sup> K<sup>-2</sup> for *p*-type Si<sub>0.95</sub>Ge<sub>0.05</sub>. Consistent results for the four applied reverse voltages are well described by the model of potential fluctuations. Moreover, the utilization of the reverse bias method is corroborated by the results obtained on Pd/*n*-type Si<sub>0.90</sub>Ge<sub>0.10</sub> SBD shown in this work.
experimental data shown in Fig. 8d were all well fitted using rather two straight lines instead of a
single straight line with a transition occurring at 140 K. Here again the values of *A*\* obtained from
these plots are in good agreement with the theoretical value of 112 A.K⁻².cm⁻² obtained for *n*-type
Si<sub>0.90</sub>Ge<sub>0.15</sub> [36, 37]. The temperature dependent reverse current-voltage characteristics of Pd/*n*--
strained type Si<sub>0.90</sub>Ge<sub>0.10</sub> structure is successfully explained on the basis of *TE* mechanism with a
Gaussian distribution of the barrier height. It is worth noting that the reverse mean values of barrier
height, deduced from Fig. 8d matches exactly with those obtained from the Φ<sub>RBn</sub> vs. 1/2k<sub>B</sub>T reported
in Fig 8c. We conclude that the *BHi* at Pd/*n*-type Si<sub>0.90</sub>Ge<sub>0.10</sub> using reverse electrical characteristics
can be satisfactorily explained by inhomogeneous model with Gaussian distribution of SBH.

IV. Conclusion

In this work, we have demonstrated that the temperature dependent reverse current-voltage characteristics can be used to investigate the barrier height inhomogeneities at Ti/*p*-type strained Si<sub>1-x</sub>Ge<sub>x</sub>Schottky contacts. The transport mode in the reverse bias is governed by the *TE* mode associated with a Gaussian SBH distribution. The abnormally temperature dependent of SBH deduced from reverse current-voltage characteristics is found to be similar to that using forward current-voltage characteristics. The decrease of homogeneous SBH Φ&#773;<sub>RBp</sub> and the increase of standard deviation σ<sub>Rs</sub> with increasing applied reverse bias are shown to be related to the increase of Schottky barrier lowering due to the image force effects and the high degree of inhomogeneity at Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> interface, respectively. The reverse bias values for Richardson constant *A*\*, extracted from the modified plots based on *BHi* model, are close to the theoretical value of 32 A cm<sup>-2</sup> K<sup>-2</sup> for *p*-type Si<sub>0.95</sub>Ge<sub>0.05</sub>. Consistent results for the four applied reverse voltages are well described by the model of potential fluctuations. Moreover, the utilization of the reverse bias method is corroborated by the results obtained on Pd/*n*-type Si<sub>0.90</sub>Ge<sub>0.10</sub> SBD shown in this work.
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"Statements and Declarations"

The authors declare that no funds, grants, or other support were received during the preparation of
this manuscript.

"Competing Interests"

The authors have no relevant financial or non-financial interests to disclose.

"Author Contributions"

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Mohammed Mamor, Khalid Bouziane, Hind Chakir and Pierre Ruterana.
All authors read and approved the final manuscript.

"Data Availability Statement"

The datasets generated during and/or analyzed during the current study are available from the
corresponding author on reasonable request.
"Statements and Declarations"

The authors declare that no funds, grants, or other support were received during the preparation of
this manuscript.

"Competing Interests"

The authors have no relevant financial or non-financial interests to disclose.

"Author Contributions"

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Mohammed Mamor, Khalid Bouziane, Hind Chakir and Pierre Ruterana.
All authors read and approved the final manuscript.

"Data Availability Statement"

The datasets generated during and/or analyzed during the current study are available from the
corresponding author on reasonable request.
<table><caption>TABLE I. Experimental data for Ti/p-strained Si<sub>0.95</sub>Ge<sub>0.05</sub>/Si SBD extracted from both reverse bias *I*<sub>R</sub>–*V*<sub>R</sub> and forward bias *I*<sub>F</sub>–*V*<sub>F</sub> curves.</caption><thead><tr><th>Bias<br>(V)</th><th>Φ<sub>RBp</sub> (V<sub>R</sub>)<br>Reverse I-V (eV)</th><th>ΔΦ<sub>im</sub><br>(meV)</th><th>Φ<sub>FBp</sub> = Φ<sub>RBp</sub> + ΔΦ<sub>im</sub></th><th>σ<sub>Rs</sub><br>(mV)</th><th>A<sup>*</sup><sup>(b)</sup><br>A.cm<sup>-2</sup>.K<sup>-2</sup></th><th>A<sup>*</sup><sup>(c)</sup><br>A.cm<sup>-2</sup>.K<sup>-2</sup></th><th>Φ<sub>RBp</sub><sup>(c)</sup><br>(eV)</th></tr></thead><tbody><tr><td>0</td><td>0.546±0.005<sup>(a)</sup></td><td>37<sup>(a)</sup></td><td>0.583 ±0.005<sup>(a)</sup></td><td>67.00<sup>(a)</sup></td><td>0.06<sup>(a)</sup></td><td>35.5<sup>(a)</sup></td><td>0.550±0.005<sup>(a)</sup></td></tr><tr><td>-1</td><td>0.541±0.005</td><td>50</td><td>0.591±0.005</td><td>74.43</td><td>0.04</td><td>35.1</td><td>0.538±0.005</td></tr><tr><td>-2</td><td>0.536±0.005</td><td>57</td><td>0.593 ±0.005</td><td>77.72</td><td>0.06</td><td>36.2</td><td>0.536±0.005</td></tr><tr><td>-2.5</td><td>0.530±0.005</td><td>60</td><td>0.590 ±0.005</td><td>78.10</td><td>0.06</td><td>43.4</td><td>0.535±0.005</td></tr><tr><td>-3</td><td>0.524±0.005</td><td>62</td><td>0.586 ±0.005</td><td>78.29</td><td>0.04</td><td>41.7</td><td>0.528±0.005</td></tr></tbody></table>

(a) Electrical parameters as determined only from forward *I*<sub>F</sub>–*V*<sub>F</sub> characteristics.

(b) Richardson constant *A*<sup>*</sup> as determined from conventional plot ln (*I*<sub>R</sub>/T<sup>2</sup>) vs. (1/*k*<sub>B</sub>*T*).

(c) Richardson constant *A*<sup>*</sup> and Φ<sub>RBp</sub> as determined from modified plot ln (I<sub>R</sub>/T<sup>2</sup>) – (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/2k<sub>B</sub><sup>2</sup>T<sup>2</sup>) vs. (1/*k*<sub>B</sub>*T*).

<table><caption>TABLE II. Experimental data for Pd/*n*-strained Si<sub>0.90</sub>Ge<sub>0.10</sub>/Si SBD extracted from both reverse bias *I*<sub>R</sub>–*V*<sub>R</sub> and forward bias *I*<sub>F</sub>–*V*<sub>F</sub> curves.</caption><thead><tr><th>Bias<br>(V)</th><th>Φ<sub>RBp</sub> (V<sub>R</sub>)<br>Reverse I-V (eV)</th><th>ΔΦ<sub>im</sub><br>(meV)</th><th>Φ<sub>FBp</sub> = Φ<sub>RBp</sub> + ΔΦ<sub>im</sub></th><th>σ<sub>Rs</sub><br>(mV)</th><th>A<sup>*</sup><sup>(b)</sup><br>A.cm<sup>-2</sup>.K<sup>-2</sup></th><th>Φ<sub>RBp</sub><sup>(b)</sup><br>(eV)</th></tr></thead><tbody><tr><td>0</td><td>0.805±0.005<sup>(a)</sup></td><td>39<sup>(a)</sup></td><td>0.84 ±0.005<sup>(a)</sup></td><td>84<sup>(a)</sup></td><td>110<sup>(a)</sup></td><td>0.806±0.005<sup>(a)</sup></td></tr><tr><td>-1</td><td>0.796±0.005</td><td>50</td><td>0.8451±0.005</td><td>111</td><td>109</td><td>0.787±0.005</td></tr><tr><td>-2</td><td>0.766±0.005</td><td>57</td><td>0.823 ±0.005</td><td>111</td><td>100</td><td>0.758±0.005</td></tr><tr><td>-2.5</td><td>0.762±0.005</td><td>60</td><td>0.822 ±0.005</td><td>112</td><td>91</td><td>0.751±0.005</td></tr><tr><td>-3</td><td>0.761±0.005</td><td>62</td><td>0.823 ±0.005</td><td>115</td><td>127</td><td>0.744±0.005</td></tr></tbody></table>

(a) Electrical parameters as determined only from forward *I*<sub>F</sub>–*V*<sub>F</sub> characteristics.

(b) Richardson constant *A*<sup>*</sup> and Φ<sub>RBp</sub> as determined from modified plot ln (I<sub>R</sub>/T<sup>2</sup>) – (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/2k<sub>B</sub><sup>2</sup>T<sup>2</sup>) vs. (1/*k*<sub>B</sub>*T*).
<table><caption>TABLE I. Experimental data for Ti/p-strained Si<sub>0.95</sub>Ge<sub>0.05</sub>/Si SBD extracted from both reverse bias *I*<sub>R</sub>–*V*<sub>R</sub> and forward bias *I*<sub>F</sub>–*V*<sub>F</sub> curves.</caption><thead><tr><th>Bias<br>(V)</th><th>Φ<sub>RBp</sub> (V<sub>R</sub>)<br>Reverse I-V (eV)</th><th>ΔΦ<sub>im</sub><br>(meV)</th><th>Φ<sub>FBp</sub> = Φ<sub>RBp</sub> + ΔΦ<sub>im</sub></th><th>σ<sub>Rs</sub><br>(mV)</th><th>A<sup>*</sup><sup>(b)</sup><br>A.cm<sup>-2</sup>.K<sup>-2</sup></th><th>A<sup>*</sup><sup>(c)</sup><br>A.cm<sup>-2</sup>.K<sup>-2</sup></th><th>Φ<sub>RBp</sub><sup>(c)</sup><br>(eV)</th></tr></thead><tbody><tr><td>0</td><td>0.546±0.005<sup>(a)</sup></td><td>37<sup>(a)</sup></td><td>0.583 ±0.005<sup>(a)</sup></td><td>67.00<sup>(a)</sup></td><td>0.06<sup>(a)</sup></td><td>35.5<sup>(a)</sup></td><td>0.550±0.005<sup>(a)</sup></td></tr><tr><td>-1</td><td>0.541±0.005</td><td>50</td><td>0.591±0.005</td><td>74.43</td><td>0.04</td><td>35.1</td><td>0.538±0.005</td></tr><tr><td>-2</td><td>0.536±0.005</td><td>57</td><td>0.593 ±0.005</td><td>77.72</td><td>0.06</td><td>36.2</td><td>0.536±0.005</td></tr><tr><td>-2.5</td><td>0.530±0.005</td><td>60</td><td>0.590 ±0.005</td><td>78.10</td><td>0.06</td><td>43.4</td><td>0.535±0.005</td></tr><tr><td>-3</td><td>0.524±0.005</td><td>62</td><td>0.586 ±0.005</td><td>78.29</td><td>0.04</td><td>41.7</td><td>0.528±0.005</td></tr></tbody></table>

(a) Electrical parameters as determined only from forward *I*<sub>F</sub>–*V*<sub>F</sub> characteristics.

(b) Richardson constant *A*<sup>*</sup> as determined from conventional plot ln (*I*<sub>R</sub>/T<sup>2</sup>) vs. (1/*k*<sub>B</sub>*T*).

(c) Richardson constant *A*<sup>*</sup> and Φ<sub>RBp</sub> as determined from modified plot ln (I<sub>R</sub>/T<sup>2</sup>) – (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/2k<sub>B</sub><sup>2</sup>T<sup>2</sup>) vs. (1/*k*<sub>B</sub>*T*).

<table><caption>TABLE II. Experimental data for Pd/*n*-strained Si<sub>0.90</sub>Ge<sub>0.10</sub>/Si SBD extracted from both reverse bias *I*<sub>R</sub>–*V*<sub>R</sub> and forward bias *I*<sub>F</sub>–*V*<sub>F</sub> curves.</caption><thead><tr><th>Bias<br>(V)</th><th>Φ<sub>RBp</sub> (V<sub>R</sub>)<br>Reverse I-V (eV)</th><th>ΔΦ<sub>im</sub><br>(meV)</th><th>Φ<sub>FBp</sub> = Φ<sub>RBp</sub> + ΔΦ<sub>im</sub></th><th>σ<sub>Rs</sub><br>(mV)</th><th>A<sup>*</sup><sup>(b)</sup><br>A.cm<sup>-2</sup>.K<sup>-2</sup></th><th>Φ<sub>RBp</sub><sup>(b)</sup><br>(eV)</th></tr></thead><tbody><tr><td>0</td><td>0.805±0.005<sup>(a)</sup></td><td>39<sup>(a)</sup></td><td>0.84 ±0.005<sup>(a)</sup></td><td>84<sup>(a)</sup></td><td>110<sup>(a)</sup></td><td>0.806±0.005<sup>(a)</sup></td></tr><tr><td>-1</td><td>0.796±0.005</td><td>50</td><td>0.8451±0.005</td><td>111</td><td>109</td><td>0.787±0.005</td></tr><tr><td>-2</td><td>0.766±0.005</td><td>57</td><td>0.823 ±0.005</td><td>111</td><td>100</td><td>0.758±0.005</td></tr><tr><td>-2.5</td><td>0.762±0.005</td><td>60</td><td>0.822 ±0.005</td><td>112</td><td>91</td><td>0.751±0.005</td></tr><tr><td>-3</td><td>0.761±0.005</td><td>62</td><td>0.823 ±0.005</td><td>115</td><td>127</td><td>0.744±0.005</td></tr></tbody></table>

(a) Electrical parameters as determined only from forward *I*<sub>F</sub>–*V*<sub>F</sub> characteristics.

(b) Richardson constant *A*<sup>*</sup> and Φ<sub>RBp</sub> as determined from modified plot ln (I<sub>R</sub>/T<sup>2</sup>) – (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/2k<sub>B</sub><sup>2</sup>T<sup>2</sup>) vs. (1/*k*<sub>B</sub>*T*).
FIGURE CAPTIONS

**Fig.g.1.** (a) A SEM image of Ti circular Schottky contacts. (b) A Schematic cross section view of the Schottky barrier diode.

**Fig. 2.** (Color online) Experimental reverse and forward *I*-*V* curves of Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD at different temperatures.

**Fig. 3.** (Color online) Variation of the effective reverse barrier heights Φ<sub>RBp</sub> as a function of temperature of Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD for different applied reverse bias. For comparison, the temperature dependence of the effective zero-bias SBH Φ<sub>0Bp</sub> as deduced from *I*<sub>F</sub>-*V*<sub>F</sub> characteristics is also shown.

**Fig. 4.** (Color online) The effective reverse barrier heights Φ<sub>RBp</sub> vs. 1/2k<sub>B</sub>T curves of Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD according to inhomogeneous model [12].

**Fig. 5.** Variation of the reverse-bias means barrier height $\bar{\Phi}_{RBp}$ and the reverse-bias standard deviation σ<sub>Rs</sub> as a function of the applied reverse bias.

**Fig. 6.** (Color online) Conventional Richardson plots ln (I<sub>R</sub>/T<sup>2</sup>) vs. (1/k<sub>B</sub>T) and modified Richardson plots ln (I<sub>R</sub>/T<sup>2</sup>) - (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/(2k<sub>B</sub><sup>2</sup>T<sup>2</sup>)) vs. (1/k<sub>B</sub>T) drawn for different applied reverse bias.

**Fig. 7.** A schematic energy band diagram of Ti/*p*-type Si<sub>1-x</sub>Ge<sub>x</sub> Schottky contacts (not to scale).

**Fig. 8.** (Color online) Experimental reverse and forward *I*-*V* curves of Pd/*n*-strained Si<sub>0.90</sub>Ge<sub>0.10</sub> SBD at different temperatures (Fig. 8 a). Fig. 8b, Fig 8c and Fig .8d are the variation of the effective reverse barrier heights Φ<sub>RBp</sub> as a function of temperature, the effective reverse barrier heights Φ<sub>RBp</sub> vs. q/2k<sub>B</sub>T curves and the modified Richardson plots ln (I<sub>R</sub>/T<sup>2</sup>) - (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/(2k<sub>B</sub><sup>2</sup>T<sup>2</sup>)) vs. (1/k<sub>B</sub>T) drawn for different applied reverse bias, respectively.
FIGURE CAPTIONS

**Fig.g.1.** (a) A SEM image of Ti circular Schottky contacts. (b) A Schematic cross section view of the Schottky barrier diode.

**Fig. 2.** (Color online) Experimental reverse and forward *I*-*V* curves of Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD at different temperatures.

**Fig. 3.** (Color online) Variation of the effective reverse barrier heights Φ<sub>RBp</sub> as a function of temperature of Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD for different applied reverse bias. For comparison, the temperature dependence of the effective zero-bias SBH Φ<sub>0Bp</sub> as deduced from I<sub>F</sub>-V<sub>F</sub> characteristics is also shown.

**Fig. 4.** (Color online) The effective reverse barrier heights Φ<sub>RBp</sub> vs. 1/2k<sub>B</sub>T curves of Ti/*p*-strained Si<sub>0.95</sub>Ge<sub>0.05</sub> SBD according to inhomogeneous model [12].

**Fig. 5.** Variation of the reverse-bias means barrier height $\bar{\Phi}_{RBp}$ and the reverse-bias standard deviation σ<sub>Rs</sub> as a function of the applied reverse bias.

**Fig. 6.** (Color online) Conventional Richardson plots ln (I<sub>R</sub>/T<sup>2</sup>) vs. (1/k<sub>B</sub>T) and modified Richardson plots ln (I<sub>R</sub>/T<sup>2</sup>) - (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/(2k<sub>B</sub><sup>2</sup>T<sup>2</sup>)) vs. (1/k<sub>B</sub>T) drawn for different applied reverse bias.

**Fig. 7.** A schematic energy band diagram of Ti/*p*-type Si<sub>1-x</sub>Ge<sub>x</sub> Schottky contacts (not to scale).

**Fig. 8.** (Color online) Experimental reverse and forward *I*-*V* curves of Pd/*n*-strained Si<sub>0.90</sub>Ge<sub>0.10</sub> SBD at different temperatures (Fig. 8 a). Fig. 8b, Fig 8c and Fig .8d are the variation of the effective reverse barrier heights Φ<sub>RBp</sub> as a function of temperature, the effective reverse barrier heights Φ<sub>RBp</sub> vs. q/2k<sub>B</sub>T curves and the modified Richardson plots ln (I<sub>R</sub>/T<sup>2</sup>) - (q<sup>2</sup>σ<sub>RS</sub><sup>2</sup>/(2k<sub>B</sub><sup>2</sup>T<sup>2</sup>)) vs. (1/k<sub>B</sub>T) drawn for different applied reverse bias, respectively.
<figure><img src="image_2.png" /><figcaption>Fig. 1</figcaption></figure>
<figure><img src="image_2.png" /><figcaption>Fig. 1</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 6</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8</figcaption></figure>