2011 Faible Tension Faible Consommation (FTFC)
Finite Difference Numerical Solution of Poisson Equation in a Schottky Barrier diode Using Maple El Mahdi Assaid, Safae Aazou Asmaa Ibral Department of Physics Faculty of Sciences B. P. 20, El Jadida, Morocco
[email protected]
El Mustapha Feddi Department of Physics E. N. S. E. T. Rabat, Morocco
equations without approximation.
Abstract: In the present study, we determine using Maple software [1] the exact numerical solution of Poisson’s equation in a Schottky barrier junction according to three different approaches. First, we consider the simple case where the space charge zone is depleted and the doping impurities are fully ionized. Then we treat the case where the space charge zone is non-depleted and the doping impurities are fully ionized. Finally, we solve rigorously the more general case where the space charge zone is non-depleted and the doping impurities are partially ionized. We use two different methods to solve the problem. In the former one, the distance of each point to the junction is calculated as a function of its potential. In the second one, we use a finite difference scheme to solve the Poisson’s equation. The calculations may be integrated into a course on semiconductor devices to show the use of Maple capabilities in the resolution of the second order non-linear differential equation governing the potential in electronic devices.
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II. BACKGROUND EQUATIONS
Let us consider a one dimensional abrupt Schottky barrier contact at , between Gold ( ) and n-type Silicon ( ) homogeneously and non degenerately doped with an excess donors concentration The junction barrier height is defined as the potential difference between metal Fermi energy and conduction band edge :
B M
( )
Where M is the metal work function which is the potential difference between vacuum energy level and metal Fermi energy. is the electron affinity which is the potential difference between vacuum energy level and semiconductor conduction band. In this paper the calculations are performed using the following values : , M 4.8 eV and 4eV [2, 3]. Figure 1 shows the energy band diagrams of the metal and the semiconductor. Before contact, the band diagrams are aligned referring to the same vacuum energy level. After contact, the Fermi energy level becomes uniform in the whole structure which leads to a deformation of semiconductor valence and conduction bands.
Keywords: Schottky barrier, Poisson equation, Finite difference method. PACS: 73.30.+y, 73.40.Ns
I. INTRODUCTION
Schottky barrier junctions are metal-doped semiconductor contacts. They are of great importance since they are present in every semiconductor device. A good understanding of Schottky barrier junctions physics is necessary for the study of other electronic and optoelectronic devices such as metalsemiconductor FET, metal-oxide-semiconductor FET, metal-semiconductor-metal photodetector. In this paper we present an electrostatic study of a Schottky barrier contact. We use Maple to determine, via two different methods, the numerical solution of the Poisson’s equation. The paper shows to young researchers and to engineering students how they can deal with semiconductor devices, and solve relative
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In semiconductor device physics, Poisson’s equation is the key equation whose resolution leads to the determination of potential, electrostatic field and electrons and holes concentrations profiles. In a general case, Poisson’s equation writes :
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( )
(
)
C. NON-DEPLETED SPACE CHARGE ZONE AND DONORS PARTIAL IONZATION
( )
In this case, the electrons concentration is not identically zero in the space charge zone and the ionized donors concentration is smaller than the doping concentration . The expression of the total charge becomes [3] :
is the total charge density. is the semiconductor dielectric constant. is the absolute value of the electron elementary charge. and are, respectively, the electrons and holes concentrations. and are, respectively, the ionized donors and acceptors concentrations.
( )
A. SPACE CHARGE ZONE FULL DEPLETION AND DONORS FULL IONIZATION APPROXIMATION
(
(
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( )
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(
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AND
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NUMERICAL
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, equation (2) reduces to : (
,
(
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First of all, we use the equality ( ). Then we integrate the differential equation and take into account the fact that far from the junction, the electric field vanish. The calculations in the case of equation (7) leads to the following equation :
is the intrinsic carriers concentration. By setting : , , and (
and
In this section, we expose the two methods used to solve equation (7) and (10). In the first method, we derive an analytic expression of the coordinate corresponding to a potential value lying between 0 and the diffusion potential .
In this case, the ionized donors concentration is equal to the doping concentration , and the electrons and holes concentrations and are not identically zero, they may be expressed as functions of the potential : (
( )
))
Equations (7) and (10) are second order non linear ordinary differential equations. They may be written as follows :
B. NON-DEPLETED SPACE CHARGE ZONE AND DONORS FULL IONZATION
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(
III. ANALYTICAL RESOLUTION
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Integrating equation (4) and setting gives the potential expression :
)
equation (9) reduces to :
at
)
(
(
By setting :
( )
( )
(
where is the diffusion tension which is the potential difference between the surface and the volume of the semiconductor. As a consequence, equation (2) writes :
In the framework of this approximation, the charge density outside the depletion region is identically zero, and the carriers concentrations and inside the depletion region are negligible compared to the ionized donors concentration. Moreover, the ionized donors concentration is equal to the doping concentration . So, the Poisson’s equation for , where stands for the space charge zone width, reduces to :
Integrating equation (3) and setting ( ) leads to the electric field expression :
(
(
)
(
) (
)
At this level of calculation, we emphasize that for , ( ) and . By setting
( )
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( ) , the equation giving the coordinate of each potential value lying between 0 and writes :
Next, we make the difference between equations ( ) and ( ), take into account the two equations describing the potential continuity at the device edges and and lead to a set of equations. The first equation is :
( )
√ ( )
∫ √
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( ) (
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In figure 2, the potential
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(
(
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) (
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In figure 3, the potential ( ), calculated via the finite difference scheme according to the three approaches detailed above, is plotted versus the distance to the metal-semiconductor junction. IV. CONCLUSION
In this work, Maple computer algebra software [1] was used to determine the exact numerical solution of Poisson’s equation in a Schottky barrier junction according to three different approaches. First, the simple case where the space charge zone is depleted and the doping impurities are fully ionized was examined. Then, the more realistic case where the space charge zone is non-depleted and the doping impurities are fully ionized was treated. Finally, the more
( ) (
( ) (
( ))
The equations above correspond to a tridiagonal matrix problem for the error vector . The algorithm we pursue to solve this problem is presented in reference [4]. First, we give the ( )⁄( ). Then initial vector we solve the linear system ( ) for and update our original values via . The algorithm stops when the error becomes small in comparison to a predefined tolerance.
where is lying between and and is the stepsize of the grid. Then we multiply ⁄ and linearize them by equations ( ) by making the substitutions : , and , where is a guess value for and is the error in that guess. As a consequence, we obtain a new set of equations :
( )
)
(
In the second method, we use a finite difference scheme to solve equation (11). We assume that the potentials at the edges of the device are known. These potentials are at and at is the length of the Schottky barrier diode. To obtain a numerical solution of equation (11), we use a uniform discrete grid. applying equation (11) at every grid point leads to a set of non linear equations : (
(
Finally, the last equation is expressed as :
( ),
calculated via the first method according to the three approaches detailed above, is drawn as a function of the depth in the semiconductor.
( )
)
The general equation writes :
( )
∫
))
)
To solve equation (10), we use the same procedure. The analytic calculations lead to the equation giving the coordinate of each potential value lying between 0 and :
√
(
)
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general case where the space charge zone is nondepleted and the doping impurities are partially ionized was rigorously solved. Two different methods were used. In the former one, the distance of each point to the junction is expressed analytically as a function of its potential. In the second one, a finite difference scheme is pursued to solve the Poisson’s equation. The numerical results may be used to calculate the electric field, the carriers concentrations profiles and the current in the junction. The paper emphasized the use of Maple capabilities in the resolution of the second order non-linear differential equation governing the potential in electronic devices.
(in ) of a Schottky barrier junction as a function of the coordinate (in ).
REFERENCES
Figure 2 : The Potential (in ), calculated via the first method, versus the distance to the junction (in ) according to the first approach (blue diamonds) the second approach (green boxes) and the third approach (red circles).
[1] Maple V Release 4, Copyright 1981-1995, Waterloo Maple Inc. [2] B. Van Zeghbroeck, Principles of semiconductor devices, available online at the electronic address : http://ecee.colorado.edu/~bart/book/book/index. html [3] H. Mathieu, T. Bretagnon and P. Lefebvre, Physique des semiconducteurs et des composants électroniques, Dunod, Paris, 2001. [4] S. Seahra, Non linear Boundary value problems (Relaxation methods), Maple worksheet available online at the electronic address : http://www.math.unb.ca/~seahra/teaching/4503/
Figure 3 : The Potential , calculated via the finite difference, drawn against the distance to the junction (in ) according the first approach (blue diamonds) the second approach (green boxes) and the third approach (red circles).
FIGURES
Figure 1 : Vacuum energy level (brown), conduction band (red), valence band (green) and Fermi energy
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