ON THE QUANTUM CORRECTION IN THE HYDRODYNAMIC MODELS FOR ELECTRONIC TRANSPORT IN NANOSCALE SEMICONDUCTOR DEVICES Eugenia Tulcan-Paulescu1, Dan Comănescu2∗ 1
West University of Timişoara, Faculty of Physics, Bd. V. Pârvan no. 4, 300223, Timişoara, Romania, e-mail:
[email protected], fax. +40256592383 2 West University of Timişoara, Faculty of Mathematics and Computer Science, Department of Mathematics, Bd. V. Pârvan no. 4, 300223, Timişoara, Romania, e-mail:
[email protected], tel:+40256592281, fax. +40256592316
Abstract: In this paper we present in a comparative way some classical and quantum hydrodynamic models for electronic transport in semiconductor devices. The Quantum Hydrodynamic Model (QHD) shows a remarkable potential to describe the nanostructured devices, as quantum well solar cells. The main objective of this study is to assess the quantum Bohm potential influence on the equations solutions and subsequently on the carrier transport. Results of numerical simulations concerning device which incorporate various nanostructures are presented. Keywords: quantum hydrodynamics; ballistic transport, differential equations PACS: 85.30.De, 85.30.Mn MSC: 76Y05
1. Introduction The modern industry relies heavily on the use of semiconductor devices. Nowadays, semiconductor materials are contained in almost all electronic devices. Important examples of semiconductor devices are quantum devices that are based on quantum mechanical phenomena. A significant problem is to describe the electron flow through a semiconductor device due to the application of a voltage. Depending on the size of the semiconductor device and other physical aspects, there are several different models describing the flow. The studied problem is applicable in the case of the ballistic transport of the carriers in symmetrical devices n+-n-n+. At work temperature it is assumed that the impurity atoms are completely ionized, which means that in the Poisson equation the doping concentration C is added to the electron distribution. It has to be mentioned that in this situation the impurity ions are positive. This kind of doping creates an electrical field, which eliminates the electrons from the central place (middle) of the device having as a result a region with fewer carriers. The equations of the zero-temperature quantum hydrodynamic model (QHD) consists of the conservation laws of mass moment for the particle density n, the particle current density and the Poisson's equation for the electric potential V (see [5]):
(1)
∗
Corresponding author:
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Here Ω is the domain occupied by the semiconductor, (0,T*) is the temporal interval, C denote the doping profile of the semiconductor, respectively ε and λ denote the scaled Planck constant and the scaled Debye length. The quantum term, the so-called Bohm potential, is ε2 ∆x n . Neglecting the quantum term one obtain the hydrodynamic equations (HD). 2 n In this paper we study the 1-dimensional and stationary case. In this situation we have Ω=[0,L] with L the device diameter. It is easy to see that in the both 1-dimensional and stationary models the current density J is a constant. We study the case in that J=0. In our hypotheses the equations of the zero-temperature quantum hydrodynamic model (QHD) and hydrodynamic model (HD) are:
(2)
d2 n 2 2 d V + ε d d x = 0 − (b) (a ) d x 2 d x n 2 d 2V = n − C, λ 2 d x
dV d x = 0 d 2V λ2 = n − C, d x 2
The equations are supplemented with boundary conditions. The total charge C-n vanishes at the boundary: n(0)=C(0), n(L)=C(L). In this paper the doping profile is symmetric: C(x)=C(L-x) for all x in [0,L]. The main objectives of this paper are: - To describe the Hydrodynamic equations and Quantum Hydrodynamic Model in our case of interests. - To present the mathematical method used. - To design for some particular cases the particle density n and the electric potential V. - To underline that the quantum aspects are suggestive.
2. The study of Hydrodynamic Model It is easy to see the following result. THEOREM 1. Let (n,V) a solution of (2.b) with n>0 then we have:
n( x) = C ( x) and V ( x) = V0 ∀x ∈ [0, L] with V0 a real constant.
3. The study of symmetric solutions of Quantum Hydrodynamic Model The first equation of problem (2.a) implies: (3) where q is a real constant. Using the change of variables: (4) and the expression of V we obtain the forth-order differential equation: (5)
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In this paper we study the symmetric solutions of this equation; the solutions which verifies the relation w( x) = w( L − x) for all x ∈ [0, L]. In this situation it is sufficient to study the function w on the interval [ L / 2, L]. If the total charge C-n vanishes at the boundary then we obtain: w( L) = C ( L) . (6) By symmetry of the function w we have the relations:
dw L d 3w L ( ) = 0, ( ) = 0. dx 2 dx 3 2
(7)
Supplementary we consider that the second-derivative of particle density n has the same value on L/2 in the classical model HD and in the QHD model. We obtain:
1 d 2C L L d 2w L ( ) ( ) w( ) = w0 , = 2 2 w0 dx 2 2 dx 2 2
(8)
The main mathematical problem is to find the solutions w of the Cauchy problem with the differential equation (5) and initial conditions of the form (7) and (8) having the properties: - w is defined on the interval [ L / 2, L] ; - is verified the relation (7). Our method of investigation of the particle density n and electric potential V is: - To determine (numerically) a solution w of the main mathematical problem; - To find (numerically) the particle density n with the relation (4); - To find (numerically) the electric potential V using (3). We consider the scaled parameters L = 1, ε = 0.114, λ = 0.1 and a set of four doping profiles (see Figure 1): (9) C i ( x) = 1 + 0.45[tanh(10(10 x − 5 − i )) − tanh(10(10 x − 5 + i ))], x ∈ [0,1], i ∈ {1,2,3,4}. To solve the main mathematical problem we use the software package MAPLE 9.5. We find a numerical solution using a Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant. For the electric potential V we consider the boundary conditions V(0)=V(1)=0. In the figures 2, 3, 4 and 5 are represented the particle density n and electric potential V in QHD model (gray) and HD model (black). The Figure k ( k ∈ {2,3,4,5} ) is corresponding to the doping profile Ck-1. 4. Conclusions In the case of a macroscopic device - for which the potential variation is of tens nanometer order - the quantum term Bohm becomes much smaller compared with others and the system has a classical behavior. As the dimension of the potential barrier becomes smaller the influence of the quantum term increases and causes a clear difference between the two concentration profiles. A larger width of the doping region n leads to fewer carriers. On the other hand, a narrow barrier exhibits a net concentration of carriers caused by tunneling. The tunneling takes place symmetrically from both sides. It has to be mentioned that the region with less carriers is divided in two distinct domains on both sides of the undoped region.
Acknowledgements The authors acknowledge support from CEEX Program 247 / 1 / 2006.
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References [1] Dong J., Mixed boundary-value problems for quantum hydrodynamic models with semiconductors in thermal equilibrium, Electronic Journal of Differential Equations, Vol 2005, No. 123, pp. 1-8, (2005) [2] Dreher M., Solutions to the equations of viscous quantum hydrodynamics in multiple dimensions, Konstanzer Schriften in Mathematik ind Informatik, Nr. 215, Mai (2006). [3] Gamba I.M., Jüngel A., Positive solutions to the singular second and third order differential equations for quantum fluids, Arch. Ration. Mech. Anal., 156 , pp. 183-203, (2001) [4] Jüngel A., Milišić J-P., Macroscopic Quantum Models with and without collisions, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), Vol. 2, No. 2, pp. 251-279, (2007) [5] Jüngel A., Tang S., Numerical approximation of the viscous quantum hydrodynamic model for semiconductors, Applied Numerical Mathematics, vol. 56, Issue 7 (July), pp. 899-915, (2006) [6] Jüngel A., Mariani M. C., Rial D., Local Existence of Solutions to the Transient Quantum Hydrodynamic Equations, Konstanzer Schriften in Mathematik ind Informatik, Nr. 145, Mai (2001).
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