# Optimum Barrier Height for SiC Schottky Barrier Diode

Authors: Alaa El-Din Sayed Hafez; Mohamed Abd El-Latif
Journal: ISRN Electronics
 (2013)

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## Abstract

The study of barrier height control and optimization for Schottky barrier diode (SBD) from its physical parameters have been introduced using particle swarm optimization (PSO) algorithm. SBD is the rectifying barrier for electrical conduction across the metal semiconductor (MS) junction and, therefore, is of vital importance to the successful operation of any semiconductor device. 4H-SiC is used as a semiconductor material for its good electrical characteristics with high-power semiconductor devices applications. Six physical parameters are considered during the optimization process, that is, device metal, mobile charge density, fixed oxide charge density, interface trapped charge density, oxide thickness, and voltage drop across the metal-semiconductor contact. The optimization process was performed using a MATLAB program. The results show that the SBD barrier height has been optimized to achieve a maximum or minimum barrier height across the contact, in addition to the ability of controlling the physical parameters to adjust the device barrier height.


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## Body

## 1. Introduction

The successful operation of any semiconductor device is of vital importance. The barrier height reflects the mismatch in the energy position of semiconductor majority carrier band edge and the metal Fermi level across the MS interface. The ability to control the magnitude of this barrier height is crucial for the advancement of future electronic devices to higher functionality and smaller physical dimensions. Several researches in the literature studied the barrier height control during manufacturing process [1–5]. SiC material offers superior material properties such as large breakdown electric field, large band gap, high electron saturated velocity, and high thermal conductivity. This makes it a viable semiconductor for high-voltage application and high-temperature operation with reduced power loss. 4H-SiC is used for its advantage with high-power applications [6–8]. This paper proposes an optimization technique for controlling and optimizing the SBD barrier height to achieve the required device physical parameters during the design time.


## 2. Theoretical Description

The structure of Schottky barrier diode (SBD) is illustrated schematically in Figure1. In SiC, the device regions, where electrons and holes behave as bulk carriers, the Poisson equation can be described as follows [9]:
(1)d2V(x)dx2=qε(NA-(x)-ND+(x)+n(x)-p(x)),
where (p) and (n) are the densities of holes and electrons and ND, NA are the densities of ionized donors and acceptors; the holes and electrons densities are [9]
(2)p(0)=ppoexp(-βψs),n(0)=npoexp(βψs).
The electric field at the surface is
(3)ξs=2kTqLDF(βψs,npoppo).
The space charge per unit area required to produce this field is
(4)Qsc=-εsξs=2εskTqLDF(βψs,npoppo).Figure 1
Energy band diagram ofp-type SBD.Barrier Height Calculation. The voltage drop across the oxide layer can be obtained by the application of Gauss’s law to the metal and semiconductor surfaces as follows:
(5)ΔVi=diεiQt,
where εi is the permittivity of the interfacial layer, di is the thickness of interfacial layer, and Qt is described as
(6)Qt=Qss+Qsc+Qm+Qf,
where Qss is the surface states charge density, Qsc is the space charge density, Qm is the mobile charge density, and Qf is the fixed oxide charge density [10]. Consider
(7)Qss=-q2Dit(ϕB+ϕs+ϕo),
where Dit is the interface trapped charge density, ϕB is the barrier height, ϕs is image force barrier lowering, and ϕo is the energy level at the surface. Consider
(8)Qf=q×Nf,Qm=q×Nm,
where Nf is the density of the fixed oxide charge and Nm is the density of the mobile charge. Another relation of ΔV can be obtained by inspection of the energy band diagram in Figure 1. Consider
(9)ΔVi=Eg-ϕm+χs-ψs-vp-V.
The device surface potential ψs can be easily calculated; the barrier height of SBD can be described as follows:
(10)ϕB=ψs+vp.


## 3. Optimization Process

PSO is a population-based stochastic optimization technique [11–14], inspired by social behaviour of bird flocking or fish schooling. PSO shares many similarities with evolutionary computation techniques such as genetic algorithm (GA) [15]. The system is initialized with a population of random solutions and searches for optima by updating generations. However, unlike GA, PSO has no evolution operators such as crossover and mutation. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum particles. It requires only primitive mathematical operators and is computationally inexpensive in terms of both memory requirements and speed. In reality, PSO and GA techniques are too similar, and by making some changes to GA, you have your PSO algorithm. At the beginning, the PSO algorithm randomly initializes a population (called swarm) of individuals (called particles). Each particle represents group of device physical parameters. The particles evaluate their position relative to a goal of the iteration. In each iteration, every particle adjusts its trajectory (by its velocity) toward its own previous best position, and toward the previous best position attained by any member of its topological neighbourhood. If any particle’s position is close enough to the goal function, it is considered as having found the global optimum and the recurrence is ended. In our optimization, 40 particles are used in PSO, which is a balance between the accuracy required in searching for the global optimum and time consumed. This procedure, whose flowchart is shown in Figure 2, iterates a predefined number of consecutive particles. The iteration particle is updated by the following two “best” values. The first one is the best solution (fitness) it has achieved so far. (The fitness value is also stored.) This value is called pbest. Another “best” value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the population. This best value is a global best and called gbest. When a particle takes part of the population as its topological neighbours, the best value is a local best and is called lbest. After finding the two best values, the particle updates its velocity and positions as follows:
(11)v=v+c1·rand·(pbest-present)+c2·rand·(gbest-present)present=present+v,
where v and is the particle velocity present is the current particle (solution). pbest and gbest are defined as stated before. rand is a random number between 0 and 1. c1,  c2 are learning factors. Usually, c1=c2=2 [12].Figure 2
PSO flowchart.The device physical parameters used in the optimization process and its limits values are demonstrated in Table1. The charges’ range is indicated in Figure 3. The metals used and its corresponding metal work function values are illustrated in Table 2.Table 1
Device physical parameters during the optimization process.
Parameter
Extended range
Compressed range
Comments
ϕ
m
3.5 : 5.1 eV
3.5 : 5.1 eV
For available metals
N
A
1
E
14 : 1E16
1
E
15 : 1E16
N
m
1
E
10 : 1E14
1
E
12 : 1E13
N
f
1
E
10 : 1E14
1
E
12 : 1E13
D
it
1
E
10 : 1E14
1
E
12 : 1E13
d
ox
1
E
-
8 : 1.5E-7
5
E
-
8 : 1.1E-7Table 2
Metal used and corresponding metal work function.
Metal
Sc
Al
Ti
Mo
Ca
ϕ
m (eV)
3.5
4.1
4.4
4.69
5.1Figure 3
The range of charges used during the optimization process.


## 4. Optimization Results

Optimization process is divided into three sections. The first section is a complete optimization of the entire extended range of physical parameters to achieve a maximum or minimum SBD barrier height. The second optimization fixes the applied voltage parameter to minimum value (0)V and maximum value (1)  V with the entire range of metals. The third optimization used a compressed range of physical parameters for each metal in the entire range of the metal work function. Table 3 shows the optimization results of the first section, which indicates the device physical parameters to achieve a maximum and minimum SBD barrier height. Optimization process is performed with constant applied voltages and added separately maximization for each metal in the entire band, which is indicated in Table 4. Table 5 illustrates the results of the third section which include the optimization process for each metal with constant voltage values and using the compressed range of the physical parameters. The demonstrated results show that these optimization procedures can control, maximize, or minimize the device barrier height to specific value related to the application requirements.Table 3
Optimization results for the first section.
Results
V
ϕ
m (eV)
N
A (cm−3)
N
m (cm−3)
N
f (cm−3)
D
it (cm−3)
d
ox (cm)
ϕ
B (V)
Maximize
0.7
4.4
4
E
15
4
E
12
8
E
13
6
E
11
9
E
-
8
3.328
Minimize
0.6
4.1
4
E
15
8
E
10
8
E
12
4
E
13
2.5
E
-
7
0.028Table 4
Optimization results for the second section.
Results
V
ϕ
m (eV)
N
A (cm−3)
N
m (cm−3)
N
f (cm−3)
D
it (cm−3)
d
ox (cm)
ϕ
B (V)
Maximize
0
3.5
8
E
14
6
E
11
2
E
11
8
E
13
5
E
-
8
3.328
Minimize
0
4.4
4
E
15
2
E
11
2
E
11
8
E
13
2.5
E
-
7
0.029
Maximize
1
5.1
6
E
14
4
E
10
4
E
10
1
E
12
1.5
E
-
7
3.328
Minimize
1
4.4
8
E
10
8
E
10
1
E
13
4
E
13
1.5
E
-
7
0.028
Maximize
0
3.5
1
E
15
1
E
12
8
E
13
6
E
11
2
E
-
7
3.328
Maximize
0
4.1
8
E
15
4
E
13
2
E
11
1
E
12
3
E
-
7
3.328
Maximize
0
4.4
6
E
15
8
E
10
6
E
11
1
E
13
2
E
-
7
3.328
Maximize
0
4.69
4
E
15
1
E
12
4
E
10
4
E
13
2
E
-
7
3.328
Maximize
0
5.1
1
E
15
4
E
13
1
E
12
8
E
12
1.5
E
-
7
3.328
Maximize
1
3.5
6
E
14
2
E
11
8
E
13
4
E
10
1
E
-
8
3.328
Maximize
1
4.1
1
E
16
4
E
12
8
E
13
6
E
11
9
E
-
8
3.328
Maximize
1
4.4
1
E
15
8
E
13
1
E
12
2
E
11
7
E
-
8
3.328
Maximize
1
4.69
8
E
15
1
E
13
4
E
13
1
E
12
1
E
-
8
3.328
Maximize
1
5.1
6
E
15
8
E
13
6
E
11
8
E
10
7
E
-
8
3.328Table 5
Optimization results for the third section.
Results
V
ϕ
m (eV)
N
A (cm−3)
N
m (cm−3)
N
f (cm−3)
D
it (cm−3)
d
ox (cm)
ϕ
B (V)
Maximize
0
3.5
7
E
15
4
E
12
1
E
12
1
E
13
7.5
E
-
7
3.328
Minimize
0
3.5
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
2.42
Maximize
0
4.1
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
2.94
Minimize
0
4.1
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
1.79
Maximize
0
4.4
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
2.92
Minimize
0
4.4
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
1.47
Maximize
0
4.69
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
2.9
Minimize
0
4.69
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
1.16
Maximize
0
5.1
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
2.83
Minimize
0
5.1
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
0.73
Maximize
1
3.5
6
E
15
2
E
12
3
E
12
9
E
12
9.5
E
-
8
3.32
Minimize
1
3.5
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
137
Maximize
1
4.1
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
3.09
Minimize
1
4.1
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
0.73
Maximize
1
4.4
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
2.47
Minimize
1
4.4
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
0.42
Maximize
1
4.69
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
1.88
Minimize
1
4.69
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
0.028
Maximize
1
5.1
1
E
16
1
E
12
1
E
12
1
E
13
1.1
E
-
7
3.328
Minimize
1
5.1
1
E
15
1
E
13
1
E
13
1
E
12
1.1
E
-
7
0.028


## 5. Conclusion

The barrier height for 4H-SiC SBD has been calculated from its device physical parameters using an iterative solution. This calculation includes the effect of the oxide charge density in the interfacial device layer, oxide thickness, metal type, doping concentration of 4H-SiC, and the external applied voltage. The device physical parameters are optimized using PSO to achieve a maximum or minimum barrier height with several conditions. The paper concluded that the SBD barrier height can be controlled by changing the metal type and the concentration range of both the oxide charges and thickness.



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*Source: 528094-2013-07-31.xml*