Distribution Of Barrier Heights Research Articles

Multi-Gaussian distribution of barrier height in diamond-like carbon interfacial-layered Schottky devices

Pieces of information about the physical and electronic properties of diamond-like carbon (DLC) interfacial-layered Schottky devices are crucial because DLC is known for its durability against harsh conditions such as high voltage, high temperature, and radiative environments. Therefore, this study focused on determining some critical properties of DCL interlayered Schottky devices, such as current-conduction mechanisms (CCMs) and the shape of the barrier height of the device. Some graphics, such as n-ΦB0vs T, ΦB0vs n, ΦB0vs q/2kT, 1/n−1vs q/2kT, and ln(I0/T2-(qσs)/2k2T2vs kT/q were obtained from the temperature-dependent current-voltage (I–V-T) data to determine the shape of the barrier height (BH) and to understand CCMs of this MIS-type device. Obtained results revealed that the device exhibited different behaviours in three different temperature regions: 80–170 K, 200–290 K and 320–410 K, which were called Low Temperatures (LTs), Moderate Temperatures (MTs) and High Temperatures (HTs), respectively. It was also observed that all these graphs exhibited linear behaviour separately for these three temperature regions. Therefore, these results showed that the barrier shape of this DLC interlayered Schottky device is not homogeneous, and it has a Multi-Gaussian distribution due to three different linear behaviours. On the other hand, in addition to the Thermionic Emission (TE) mechanism, it was also understood that Field Emission (FE) and Thermionic Field Emission (TFE) mechanisms, known as Quantum Mechanical Tunnelling (QMT) mechanisms, were effective current conduction mechanisms, especially at low and moderate temperatures for this device.

Analysis of temperature-dependent current–voltage characteristics of Schottky diodes by the modified thermionic emission current model

We have investigated the behavior of current flow across an inhomogeneous Schottky diode (SD) as a function of temperature by numerical simulation. We have used the modified thermionic emission (TE) current expression with a Gaussian distribution of potential barrier heights. This modified TE model assumes the presence of a series of low-barrier patches at the Schottky contact and semiconductor interface. First, we have discussed the behavior of the patch current compound relative to the TE compound in the inhomogeneous SD at 300, 200, and 100 K, as a function of standard deviation and the number of circular patches N. Then, we have investigated the behavior of temperature- and bias-dependent and bias-independent current vs voltage (I–V–T) characteristics in the 75–300 K range. In bias-dependent I–V–T curves obtained for σ1=4.35×10−5cm2/3V1/3 and σ2=7.35×10−5cm2/3V1/3 at N1=1.81×106 or N2=1.81×108, an intersection behavior has been observed in the I–V curve at 75 K for σ2 at both N values; however, the same behavior has been not observed for σ1 at both N values due to σ1<σ2. That is, the current for σ2 at 75 K has exceeded the current at higher temperatures. This behavior has been ascribed to the effective BH to decrease with decreasing temperature value. In the I–V–T curves independent of bias, such an intersection has not been observed for σ1 while it has been observed for σ2 in the I–V curves at both 75 and 100 K. Thus, it has been concluded that the bias-dependeσnt I–V equations must be used to avoid this intersection behavior while fitting the experimental I–V curve of an SD to the theoretical I–V curve.

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

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 Si0.95Ge0.05 Schottky barrier diodes (SBD) has been investigated using reverse current–voltage-temperature (IR-VR-T) characteristics over the temperature range of 100–300 K. The temperature dependence of the measured barrier heights (ΦRBp) 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 (Ф‾RBp) and its corresponding standard deviation (σRs) were determined for different reverse voltages. A decrease in Ф‾RBp and an increase in σRs with increasing reverse bias are observed, indicating a large barrier height distribution upon high applied reverse bias. The deduced average zero-field barrier height (ФFBp≈0.59eV) from IR-VR 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−2 K−2 for our p-type strained Si0.95Ge0.05 alloy. Finally, the results obtained on Pd/n-type Si0.90Ge0.10 structure shown in this work fully support the utilization of such reverse bias method.

Schottky contacts on sulfurized silicon carbide (4H-SiC) surface

In this Letter, the effect of a sulfurization treatment carried out at 800 °C on silicon carbide (4H-SiC) surface was studied by detailed chemical, morphological, and electrical analyses. In particular, x-ray photoelectron spectroscopy confirmed sulfur (S) incorporation in the 4H-SiC surface at 800 °C, while atomic force microscopy showed that 4H-SiC surface topography is not affected by this process. Notably, an increase in the 4H-SiC electron affinity was revealed by Kelvin Probe Force Microscopy in the sulfurized sample with respect to the untreated surface. The electrical characterization of Ni/4H-SiC Schottky contacts fabricated on sulfurized 4H-SiC surfaces revealed a significant reduction (∼0.3 eV) and a narrower distribution of the average Schottky barrier height with respect to the reference untreated sample. This effect was explained in terms of a Fermi level pinning effect induced by surface S incorporation.

Creep failure of hierarchical materials

Creep failure of hierarchical materials is investigated by simulation of beam network models. Such models are idealizations of hierarchical fibrous materials where bundles of load-carrying fibers are held together by multi-level (hierarchical) cross-links. Failure of individual beams is assumed to be governed by stress-assisted thermal activation over local barriers, and beam stresses are computed by solving the global balance equations of linear and angular momentum across the network. Disorder is mimicked by a statistical distribution of barrier heights. Both initially intact samples and samples containing side notches of various length are considered. Samples with hierarchical cross-link patterns are simulated alongside reference samples where cross-links are placed randomly without hierarchical organization. The results demonstrate that hierarchical patterning may strongly increase creep strain and creep lifetime while reducing the lifetime variation. This is due to the fact that hierarchical patterning induces a failure mode that differs significantly from the standard scenario of failure by nucleation and growth of a critical crack. Characterization of this failure mode demonstrates good agreement between the present simulations and experimental findings on hierarchically patterned paper sheets.

Effects of Activation Barriers on Quenching to Stabilize Prebiotic Chemical Systems.

We have previously shown in model studies that rapid quenches of systems of monomers interacting to form polymer chains can fix nonequilibrium chemistries with some lifelike properties. We suggested that such quenching processes might have occurred at very high rates on early Earth, giving an efficient mechanism for natural sorting through enormous numbers of nonequilibrium chemistries from which the most lifelike ones could be naturally selected. However, the model used for these studies did not take account of activation barriers to polymer scission (peptide bond hydrolysis in the case of proteins). Such barriers are known to exist and are expected to enhance the quenching effect. Here, we introduce a modified model which takes activation barriers into account and we compare the results to data from experiments on quenched systems of amino acids. We find that the model results turn out to be sensitive to the width of the distribution of barrier heights but quite insensitive to its average value. The results of the new model are in significantly better agreement with the experiments than those found using our previous model. The new parametrization of the model only requires one new parameter and the parametrization is more physical than the previous one, providing a chemical interpretation of the parameter p in our previous models. Within the model, a characteristic temperature Tc emerges such that if the temperature of the hot stage is above Tc and the temperature of the cold stage is below it, then the 'freezing out', in a quench, of a disequilibrium ensemble of long polymers is expected. We discuss the possible relevance of this to models of the origin of life in emissions from deep ocean rifts.

InGaN based Schottky barrier solar cell: Study of the temperature dependence of electrical characteristics

The indium gallium nitride (InGaN) semiconductor alloy shows great promise for high-efficiency thin-film solar cells due to its intrinsic characteristics. However, there are major challenges in the development of high-quality p-doped and indium-rich InGaN layers to fabricate a pn or pin solar cell. This study focuses on the development of an alternative structure, not requiring p-doping, with a Schottky contact on n-type InGaN. The InGaN absorber was grown by metalorganic chemical vapor deposition and its structural, morphological and optical properties studied. Structural analysis by X-ray diffraction showed the growth of single crystal layers without phase separation. Indium compositions which vary up to around 8.3 % were confirmed by transmission measurements. The formation of V-shaped pits was observed with increased indium composition at lower growth temperatures. Schottky diodes based on InGaN with platinum metallic contacts were fabricated and their current–voltage characteristics studied in detail as a function of temperature. This analysis of the conduction mechanisms in Schottky/InGaN devices permitted in particular the extraction of the diode parameters as a function of the temperature showing a high rectification with an ideality factor of 1.15 with reverse saturation current of 6.7×10−9A at room temperature for an indium composition of 5.52 %. In addition, the analysis revealed the presence of a dual Gaussian distribution of barrier heights in a specific temperature range. Finally, the incident photon-to-electron conversion efficiency (IPCE) of the fabricated InGaN Schottky solar cells, measured using intensity-modulated photocurrent spectroscopy, reached a peak value of 51 %, which is among the highest published values for InGaN based solar cells.

(Invited) Optically and Chemically Modulated Oxygen Transport across Interfaces

Materials that conduct oxygen are central to many technologies that promise more highly efficient chemical to electrical and electrical to chemical energy conversion, as exemplified by solid oxide fuel and electrolysis cells (SOFC/SOEC), permeation membranes for partial oxidation of hydrocarbons and solar thermal driven chemical conversions. We begin by describing recent insights achieved in controlling the oxygen reduction reaction (ORR) on mixed ionic electronic conductors (MIEC). We have been able to demonstrate the ability to reversibly depress or enhance the oxygen exchange reaction at MIEC surfaces by orders of magnitude by infiltrating the surfaces with binary oxides with controlled acidity. This we demonstrate can be correlated with the ability of the surface additives to deplete or accumulate electrons needed for the ORR. With these insights, we have been able to understand more fundamentally the source of chromia-induced poisoning of SOFC cathodes and the means to recover performance following poisoning. We next turn our attention to the well known blocking of oxygen ions at grain boundaries in solid oxide electrolytes where grain boundary resistances can drop the overall oxygen ion conductance in polycrystalline solid electrolytes by orders of magnitude. The barriers to oxygen transport across grain boundaries are largely attributed to space charge barriers that form at the grain boundary due to the positive core charge of the grain boundaries. We have demonstrated the ability to modulate the barrier potentials by illumination with above band gap light, illustrating the ability to enhance ionic conduction in solid electrolytes at reduced temperatures, while at the same time demonstrating a novel method for interrogating the distribution of barrier heights at grain boundaries and the dynamics associated with the resultant redistribution of oxygen vacancies in space charge regions under conditions of photo-induced electron-hole generation and recombination. This phenomenum is yet another example of the rapidly developing field of opto-ionics.

Barrier Height, Ideality Factor and Role of Inhomogeneities at the AlGaN/GaN Interface in GaN Nanowire Wrap-Gate Transistor.

It is essential to understand the barrier height, ideality factor, and role of inhomogeneities at the metal/semiconductor interfaces in nanowires for the development of next generation nanoscale devices. Here, we investigate the drain current (Ids)-gate voltage (Vgs) characteristics of GaN nanowire wrap-gate transistors (WGTs) for various gate potentials in the wide temperature range of 130-310 K. An anomalous reduction in the experimental barrier height and rise in the ideality factor with reducing the temperature have been perceived. It is noteworthy that the variations in barrier height and ideality factor are attributed to the spatial barrier inhomogeneities at the AlGaN/GaN interface in the GaN nanowire WGTs by assuming a double Gaussian distribution of barrier heights at 310-190 K (distribution 1) and 190-130 K (distribution 2). The standard deviation for distribution 2 is lower than that of distribution 1, which suggests that distribution 2 reflects more homogeneity at the AlGaN/GaN interface in the transistor's source/drain regions than distribution 1.

Investigation of temperature dependant current transport mechanism in Ag/In2O3/p-Si/Al heterojunction

Employing the pulsed laser ablation technique nano-crystalline thin films of Indium oxide (In2O3) are deposited successfully on boron doped silicon substrate. The structural and morphological parameters of the films are investigated by using X-ray diffraction and atomic force microscopic (AFM) spectroscopy, respectively. The deposited films are the crystalline nanostructures with (400) as preferred diffraction peak and grain size of 56.56 nm approximately. The computed roughness (Rq) of In2O3 film is 1.02 nm. The AFM analysis validated the deposition of homogeneous smooth In2O3 films with surface roughness of the order of nanometer. UV–vis spectroscopy was used to study the optical properties of the films. Greater than 80% transparency in visible region was exhibited by the films. The value of the optical band gap was found to be 3.89 eV. The electrical parameters dominating the mechanism of current transport in n-In2O3/p-Si heterojunction were explored by observing the temperature-dependent current–voltage (I–V) properties. In the temperature range of 80–300 K, the ideality factor and barrier height were found to be strongly temperature dependent. It was noticed that with the decrease in temperature there is increase in ideality factor whereas decrease in barrier height which confirm the interfacial imperfections and inhomogeniety in potential barrier height distributions. The temperature dependence of I–V data has revealed the existence of double Gaussian distribution of barrier heights with mean values of 0.85eV and 0.78eV with standard deviation of 0.014 V and 0.012 V respectively. The modified Richardson's plot gives the average barrier height and Richardson's coefficient as 0.86eV and 4.7 × 105 Am−2K−2 in the temperature range 300 K–180 K and 0.77eV and 5.2 × 105 Am−2K−2, respectively in the temperature range 160 K–80 K. The values of Richardson's constants are of the order of known theoretical value of 3.2 × 105 Am−2 K−2.

4H-SiC Schottky barrier diodes as radiation detectors: A role of Schottky contact area

Large-area SiC Schottky barrier diodes can significantly improve the sensitivity in radiation detection due to the increased interacting SiC volume. In this work, we tested a large SiC detector with an area of 1 × 1 cm2. Extensive electrical characterization was performed at temperatures ranging from 150 K to 390 K, demonstrating the impact of barrier inhomogeneities on the electrical performance of the diode. Forward current-voltage (I-V) measurements of the diodes revealed two distinct regions caused by Schottky barrier inhomogeneity present throughout the entire temperature range. The barrier heights in the low- and high-current forward voltage regions were extracted from Richardson plots corrected for the Gaussian distribution of barrier heights, yielding values of 1.52 eV and 1.79 eV, respectively. Deep-level transient spectroscopy (DLTS) revealed only one deep-level defect, Z1/2, with an activation energy for electron emission of 0.67 eV, which was assigned to the known carbon vacancy. The DLTS study showed no correlation between electrically active defects and barrier inhomogeneity. An excellent energy resolution of 3.2 % was measured using a large area 241Am radiation source, consistent with values for small area SiC detectors that exhibited no barrier height inhomogeneities. The impact of temperature on the alpha particle radiation response was determined within a temperature range of 200 K to 390 K.

Investigation of barrier height distributions over the gate area of Al-SiO2-Si structures

Distributions of the gate-dielectric EBG(x, y) and semiconductor-dielectric EBS(x, y) barrier height values have been determined using the photoelectric measurement method. Modified Powell-Berglund method was used to measure barrier height values. Modification of this method consisted in using a focused UV light beam of a small diameter d =0.3 mm. It was found that the EBG(x, y) distribution has a characteristic dome-like shape which corresponds with the independently determined shape of the effective contact potential difference fMS(x, y) distribution. On the other hand, the EBS(x, y) distribution is of a random character. It is shown that the EBG(x, y) distribution determines the shape of the fMS(x, y) distribution. The model of the EBG and EBS barrier height distributions over the gate area has been proposed.

Temperature-dependent Schottky diode behavior of Ni Schottky contacts to α-Ga2O3 film epitaxially grown on sapphire substrate

We fabricated Ni Schottky contacts on α-Ga2O3 epitaxial layers grown on a c-plane sapphire substrate and explored their current–voltage (I–V) characteristics dependence on temperature spanning over a wide temperature range of 100–425K. The Ni/α-Ga2O3 Schottky diode displayed excellent rectification, exhibiting a high barrier height of 1.39 eV, a low leakage current of the order 10−12 A, and a breakdown voltage of 215 V at room temperature. The reverse leakage current was confined within an order of magnitude over the entire measured temperature range, indicating the thermal stability of the diode. A large ideality factor of 1.20 attained at room temperature clearly indicates the diode behavior deviation from the ideal thermionic emission theory. The I–V characteristics revealed that the barrier height and ideality factor shows strong temperature dependence, indicating the existence of barrier inhomogeneity at the Schottky interface. An analysis of the barrier inhomogeneity using the thermionic emission with the assumption of a Gaussian barrier height distribution implies the prevailing double Gaussian barrier height distribution in the Ni/α-Ga2O3 Schottky diode with a transition at 250 K. The Richardson constant derived from the modified Richardson plot, evaluated by assuming the barrier height Gaussian distribution, was close to the theoretical Richardson constant of α-Ga2O3.

Sub-barrier fusion hindrance and absence of neutron transfer channels

The sub-barrier fusion hindrance has been observed in the domain of very low energies of astrophysical relevance. The measured fusion cross sections can be well estimated by using formula obtained by folding a Gaussian function for barrier height distribution with the expression for classical fusion cross section for fixed barrier. The parameters of the formula vary smoothly implying its usage in estimating the excitation function. In the present work, the effects of deformation and transfer on fusion reactions have been studied and the absence of neutron transfer channels has been correlated to the hindrance in heavy-ion fusion reactions.

Interaction at a distance: Xenon migration in Mb

The transport of ligands, such as NO or O2, through internal cavities is essential for the function of globular proteins, including hemoglobin, myoglobin (Mb), neuroglobin, truncated hemoglobins, or cytoglobin. For Mb, several internal cavities (Xe1 through Xe4) were observed experimentally and they were linked to ligand storage. The present work determines barriers for xenon diffusion and relative stabilization energies for the ligand in the initial and final pocket, linking a transition depending on the occupancy state of the remaining pockets from both biased and unbiased molecular dynamics simulations. It is found that the energetics of a particular ligand migration pathway may depend on the direction in which the transition is followed and the occupancy state of the other cavities. Furthermore, the barrier height for a particular transition can depend in a non-additive fashion on the occupancy of either cavity A or B or simultaneous population of both cavities, A and B. Multiple repeats for the Xe1 → Xe2 transition reveal that the activation barrier is a distribution of barrier heights rather than one single value, which is confirmed by a distribution of transition times for the same transition from unbiased simulations. Dynamic cross correlation maps demonstrate that correlated motions occur between adjacent residues or through space, residue Phe138 is found to be a gate for the Xe1 → Xe2 transition, and the volumes of the internal cavities vary along the diffusion pathway, indicating that there is dynamic communication between the ligand and the protein. These findings suggest that Mb is an allosteric protein.

Inhomogeneous heterojunction performance of Zr/diamond Schottky diode with Gaussian distribution of barrier heights for high sensitivity temperature sensor

Herein, we demonstrated the fabrication and operation of vertical diamond Schottky barrier diode (SBD) as a high-performance temperature sensor. The current transport mechanism and temperature sensor properties of Zr/p-diamond SBD were comprehensively investigated with operating temperatures up to 448 K. At room temperature, Zr/p-diamond SBD exhibits outstanding performances with a high rectification ratio of 2.28 × 1010, a low specific on-resistance of 0.5 mΩ•cm2, a large Schottky barrier height (SBH) of 1.43 eV, a low ideality factor of 1.77, and an extremely low saturation current of 4.42 × 10−22 A. The enhanced-performance with a higher forward current and a lower reverse leakage current is observed to be across a wide temperature range from 323 to 448 K. Additionally, a significant improvement of rectifying current curve with a highest rectification ratio over 1012 is obtained at 423 K. The SBH increases whereas the ideality factor decreases with temperature increasing, indicating barrier inhomogeneities at Zr/p-diamond interface. The temperature-dependent electrical transport characteristics on spatially inhomogeneous Schottky contacts can be successfully described by potential fluctuations model. The extracted original value of Richardson constant (A*) is 0.0153 A/cm2•K2, which is much lesser than theoretical value of 96 A/cm2•K2. The discrepancy of A* value from theoretical value has been well explained by thermionic emission model with Gaussian distribution of barrier inhomogeneity. The values of mean SBH and A* extracted from modified Richardson plot depending on SBH inhomogeneity are 1.76 eV and 105.51 A/cm2•K2, respectively. With a Gaussian distribution take into account, the A* value is close to the theoretical value, and confirmed the existence of SBH inhomogeneity at Zr/p-diamond interface. Furthermore, for Zr/p-diamond SBD operating as a temperature sensor, the voltage drop across the diode decreases linearly with increasing temperature at a specific forward current level, contributing to a highest sensor sensitivity of 6.8 mV/K. The reported results in this work are among the best to date in diamond SBD temperature sensor. Although not optimized for temperature sensor, the high and thermally stable rectifying barriers make Zr/p-diamond SBD a strong candidates for high-performance diamond rectifying diode and temperature sensor.

Heavy-ion fusion reactions of astrophysical interest

The hindrance in fusion of heavy-ion reactions crops up in the region of extreme sub-barrier energies. This phenomenon can be effectively analyzed using a simple diffused barrier formula derived assuming a Gaussian distribution of fusion barrier heights. Folding the Gaussian barrier distribution with the classical expression for the fusion cross section for a fixed barrier, the fusion cross section is obtained. The energy dependence of the fusion cross section provides good description to the existing data on sub-barrier heavy-ion fusion for lighter systems of astrophysical interest. Using this simple formula, an analysis has been presented from $^{16}$O + $^{18}$O to $^{12}$C + $^{198}$Pt, all of which were measured down to $<$ 10 $\mu$b. The agreement of the present analysis with the measured values is better than those calculated even from the sophisticated coupled channels calculations. The relatively smooth variation of the three parameters of this formula implies that it may be exploited to estimate the excitation function or to extrapolate cross sections for pairs of interacting nuclei which are yet to be measured. Possible extensions of the present methodology and its limitations have also been discussed.

The Barrier's Heights and Its Inhomogeneities on Diamond Silicon Interfaces.

In this work, the electrical parameters of the polycrystalline diamonds’ p-PCD/n-Si heterojunction were investigated using temperature-dependent current–voltage (I-V) characteristics. In the temperature range of 80–280 K, the ideality factor () and energy barrier height (φb) were found to be strongly temperature dependent. The φb increases with temperature rise, while the n value decreases. The observed dependencies are due to imperfections at the interface region of a heterojunction and the non-homogeneous distribution of the potential barrier heights. Values of the φb were calculated from I-V characteristics using the thermionic emission theory (TE). The plot of φb versus 1/2 kT revealed two distinct linear regions with different slopes in temperature regions of 80–170 K and 170–280 K. This indicates the existence of a double Gaussian distribution (DGD) in heterojunctions. Parameters such as mean barrier heights and standard deviations σ were obtained from the plots linearization and read out from intercepts and slopes. They take values = 1.06 eV, σ = 0.43 eV, respectively. The modified Richardson plot is drawn to show the linear behavior in these two temperature ranges, disclosing different values of the effective Richardson constants (A*).

Negative Schottky barrier height and surface inhomogeneity in n-silicon M–I–S structures

The alleviation effect on the Schottky barrier height (SBH) (ΦB) using ultrathin titanium dioxide and hafnium dioxide dielectrics in a single layer and a bilayer stack was demonstrated. ΦB in the Pt/n-Si contact was reduced from 0.53 to −0.058, 0.3, and −0.12 eV using 3 nm TiO2, 1 nm HfO2, and high-k/high-k bilayer insertion, respectively. A maximum of 122% reduction in ΦB was obtained using bilayer dielectric insertion, which is the highest ever reduction reported so far in a Schottky diode. This was achieved by effectively passivating the semiconductor surface states by HF cleaning followed by inserting an ultrathin film produced from the novel Atomic Layer Deposition (ALD) technique. The Gaussian distribution (GD) of barrier heights all over the interface has been investigated for both Metal–Semiconductor (M–S) and Metal–Insulator–Semiconductor (M–I–S) contacts. The nonlinear behavior in a conventional Richardson plot was observed with lower values of the Richardson constant (A*). We have reported the surface inhomogeneity in both M–S and M–I–S contacts through temperature dependency of diode characteristics. The standard deviation (σ) as evidence for the Gaussian distribution of barrier heights was determined using the ln(Is/T2) vs q/2kT plot. The results were validated by a modified Richardson plot where the values of A* obtained were found to be in close agreement with the known values. As the ALD technique is known for conformity and uniformity of thin films, the dielectric insertion has proved effective in mitigating the SBH. However, the inhomogeneity in both M–S and M–I–S points to the role of dipole formation at the interface.

Heavy-ion fusion cross section formula and barrier height distribution

Methods for obtaining fusion cross section formulas are discussed, especially for those that take on an empirical form. A new expression starting with $\ensuremath{\sigma}(E)=\frac{1}{E}{\ensuremath{\int}}_{{E}_{0}}^{E}\left({\ensuremath{\int}}_{{E}_{0}}^{{E}^{\ensuremath{'}}}B({E}^{\ensuremath{''}})d{E}^{\ensuremath{''}}\right)d{E}^{\ensuremath{'}}$ has been explored. Here, $B(E)$ is a reasonable, assumed function of the barrier height distribution, $\frac{{d}^{2}(\ensuremath{\sigma}E)}{d{E}^{2}}$. The resulting analytic cross section formula reproduces very well the excitation functions for many light and heavy fusion systems across wide energy ranges, when $B(E)$ is assumed to be a multi-Gaussian function. This study offers an improved determination of the fusion barrier height distribution over other numerical techniques.