Semiconductors and Dirichlet-to-Neumann maps

Semiconductors and Dirichlet-to-Neumann maps

Antonio Leitão

Department of Mathematics, Federal University of St. Catarina, 88040-900 Florianopolis, Brazil, E-mail: [e-mail]

ABSTRACT

We investigate the problem of identifying discontinuous doping profiles in semiconductor devices from data obtained by the stationary voltage-current (VC) map. The related inverse problem correspond to the inverse problem for the Dirichlet-to-Neumann (DN) map with partial data.

Mathematical subject classification: 35R30, 82D37, 35Q60.

Key words: semiconductors, inverse doping, bipolar model.

1 Introduction

In this paper we investigate the problem of identifying discontinuous doping profiles in semiconductor devices from data obtained by the VC map for the linearized stationary bipolar model (close to equilibrium). Two different methods of data acquisition are considered:

1) Current flow measurements through a contact;

2) Pointwise measurements of the current density.

The related inverse problems correspond to the inverse problem for the DN map with partial data.

We propose a framework to handle the inverse problems and analyze relevant properties of the parameter-to-output maps. Moreover, we present a numerical experiment for the case of pointwise measurements of the current density.

The paper is outlined as follows: In Section 2 we present the transient end stationary drift diffusion equations. In the Section 3 we introduce the VC map and derive the underlying model for the analysis presented in this paper, namely the linearized stationary bipolar case (close to equilibrium). Two inverse problems corresponding to different data acquisition procedures are introduced in Section 4. Some regularity properties of the related parameter-to-output maps are verified in this section. In Section 5 we present some numerical results for a level set type iterative method and pointwise measurements of the current density. This experiment indicate that a single measurement may suffices to identify the doping profile.

2 Drift diffusion equations

The transient model

The basic semiconductor device equations consist of the Poisson equation (1a), the continuity equations for electrons (1b) and holes (1c), and the current relations for electrons (1d) and holes (1e).

This system is defined in W × (0,T), where W Ì ^d (d = 1,2,3) is a domain representing the semiconductor device. Here V denotes the electrostatic potential (- ÑV is the electric field, E = |ÑV|)), n and p are the concentration of free carriers of negative charge (electrons) and positive charge (holes) respectively and J_n and J_p are the densities of the electron and the hole current respectively. D_n and D_p are the diffusion coefficients for electrons and holes respectively. µ_n and µ_p denote the mobilities of electrons and holes respectively. The positive constants e and q denote the permittivity coefficient (for silicon) and the elementary charge (see ^Appendix).

The function R has the form R = (n,p,x) (np - ) and denotes the recombination-generation rate (n_i is the intrinsic carrier density). The bandgap is relatively large for semiconductors (gap between valence and conduction band), and a significant amount of energy is necessary to transfer electrons from the valence and to the conduction band. This process is called generation of electron-hole pairs. On the other hand, the reverse process corresponds to the transfer of a conduction electron into the lower energetic valence band. This process is called recombination of electron-hole pairs. In our model these phenomena are described by the recombination-generation rate R. Frequently adopted in the literature are the Shockley Read Hall model (_SRH) and the Auger model (_AU), defined by

where C_n, C_p, t_n and t_p are positive constants (see ^Appendix).

The function C(x) models a preconcentration of ions in the crystal, so C(x) = C₊(x) - C_-(x) holds, where C₊ and C_- are concentrations of negative and positive ions respectively. In those subregions of W in which the preconcentration of negative ions predominate (P-regions), we have C(x) < 0. Analogously, we define the N-regions, where C(x) > 0 holds. The boundaries between the P-regions and N-regions (where C change sign) are called pn-junctions.

In the sequel we turn our attention to the boundary conditions. We assume the boundary ¶W of W to be divided into two nonempty disjoint parts: ¶W = È . The Dirichlet part of the boundary ¶W_D models the Ohmic contacts, where the potential V as well as the concentrations n and p are prescribed. The Neumann part ¶W_N of the boundary corresponds to insulating surfaces, thus a zero current flow and a zero electric field in the normal direction are prescribed. The Neumann boundary conditions for system (1a)-(1e) read:

Moreover, at ¶W_D ×[0,T], the following Dirichlet boundary conditions are imposed:

Here the function U(x,t) denotes the applied potential, the constant U_T represents the thermal voltage, and V_bi is given logarithmic function [4]. We shall consider the simple situation ¶W_D = G₀È G₁, which occurs, e.g., in a diode. The disjoint boundary parts G_i, i = 0,1, correspond to distinct contacts. Differences in U(x) between different segments of ¶W_D correspond to the applied bias between these two contacts. Moreover, the initial conditions n(x,0) > 0, p(x,0) > 0 have to be imposed.

The stationary model

In this paragraph we turn our attention to the stationary drift diffusion equations. We disconsider the thermal effects and assume further ¶n/¶t = ¶n/¶t = 0. Thus, the stationary drift diffusion model is derived from (1a)-(1e) in a straightforward way. Next, motivated by the Einstein relations D_n = U_T µ_n and D_p = U_T µ_p (a standard assumption about the mobilities and diffusion coefficients), one introduces the so-called Slotboom variables u and v, which are related to the original n and p variables by the formula:

For convenience, we rescale the potential and the mobilities, i.e.

It is obvious to check that the current relations now read J_n = µ_n n_i e^VÑu, J_p = -µ_p n_i e^-VÑv.

Next we write the stationary drift diffusion equations in terms of (V,u,v)

where l² : = e/(q U_T) is the Debye length of the device, d² : = n_i and the function Q is defined implicitly by the relation Q(V,u,v,x) = (n,p,x).¹

One should notice that, due to the thermal equilibrium assumption, it follows np = , and the assumption of vanishing space charge density gives n-p-C = 0, for x Î ¶W _D. This fact motivates the boundary conditions on the Dirichlet part of the boundary.

It is worth mentioning that, in a realistic model, the mobilities µ_n and µ_p usually depend on the electric field strength |ÑV|. In what follows, we assume that µ_n and µ_p are positive constants. This assumption simplifies the subsequent analysis, allowing us to concentrate on the inverse doping problems. As a matter of fact, this dependence could be incorporated in the model without changing the results described in the sequel.

Existence and uniqueness of solutions for system (5) can only be guaranteed for small applied voltages. Therefore, it is reasonable to consider, instead of this system, its linearized version around the equilibrium point U º 0. We shall return to this point in the next section, where the VC map is introduced.

3 A simplified model

In the sequel we make some simplifying assumptions on the stationary drift diffusion equations introduced in Section 2 and derive a special case which will serve as underlying model for the inverse problem investigated in Section 4.

The linearized stationary drift diffusion equations (close to equilibrium)

We begin this paragraph by introducing the thermal equilibrium assumption for the stationary drift diffusion equations. This is a previous step to derive a linearized system of stationary drift diffusion equations (close to equilibrium).

The thermal equilibrium assumption refers to the condition in which the semiconductor is not subject to external excitations, except for a uniform temperature, i.e. no voltages or electric fields are applied. It is worth noticing that, under the thermal equilibrium assumption, all externally applied potentials to the semiconductor contacts are zero (i.e. U(x) = 0). Moreover, the thermal generation is perfectly balanced by recombination (i.e. = 0).

If the applied voltage satisfies U = 0, one immediately sees that the solution of system (5a)-(5g) simplifies to (V,u,v) = (V⁰,1,1), where V⁰ solves

In the bipolar model discussed below we shall be interested in the linearized drift diffusion system at the equilibrium. Keeping this in mind, we compute the Gateaux derivative of the solution of system (5a)-(5g) with respect to the voltage U at the point U º 0 in the direction h. This directional derivative is given by the solution (, , ) of

where the function Q₀ satisfies Q₀(V⁰,x) = Q(V⁰,1,1,x).

Linearized stationary bipolar case (close to equilibrium)

In this paragraph we present a special case, which plays a key rule in the modeling inverse doping problems related to current flow measurements.

The following discussion is motivated by the stationary VC map

Here (V,u,v) is the solution of system (5) for an applied voltage U. This operator models practical experiments where voltage-current data are available, i.e. measurements of the averaged outflow current density on G₁̶W_D.

The linearized stationary bipolar case (close to equilibrium) corresponds tothe model obtained from the drift diffusion equations (5) by linearizing the VC map at U º 0. This simplification is motivated by the fact that, due to hysteresis effects for large applied voltage, the VC map can only be defined as a single-valued function in a neighborhood of U = 0. Moreover, the following simplifying assumptions are also taken into account:

A1) The electron mobility µ_n and hole mobility µ_p are constant;

A2) No recombination-generation rate is present, i.e. = 0 (or Q₀ = 0).

An immediate consequence of our assumptions is the fact that the Poisson equation and the continuity equations decouple. Indeed, from (7) we see that the Gateaux derivative of the VC map S_C at the point U = 0 in the direction h Î H^3/2(¶W_D) is given by the expression

where (, ) solve

and V⁰ is the solution of the equilibrium problem (6); see Lemma 1.

Notice that the solution of the Poisson equation can be computed a priori, since it does not depend on h. The application (0) maps the Dirichlet data for (, ) to a weighted sum of their Neumann data and can be compared with the DN map in the Electrical Impedance Tomography (EIT).

4 Inverse Problems for Semiconductors

We begin this section verifying that the stationary VC map S_C, introduced in Section 3, is well defined in a suitable neighborhood of U = 0.

Lemma 1 [[5], Proposition 3.1]. In (5), for each applied voltage U Î B_r(0) Ì H^3/2(¶W_D) with r > 0 sufficiently small, the current J · n Î H^1/2 (G₁) is uniquely defined. Furthermore, S_C: H^3/2(¶W_D) ® H^1/2(G₁) is continuous and continuously differentiable in B_r(0). Moreover, it's derivative in direction h Î H^3/2(¶W_D) is given by the operator

(0) defined in (8).

As a matter of fact, we can actually prove that, since (, ) in (9) depend continuously (in H²(W)²) on the boundary data U Î H^3/2(¶W_D), it follows from the boundedness and compactness of the trace operator g: H²(W) ® H^1/2(G₁) that (0) is a bounded and compact operator.

Lemma 1 establishes a basic property to consider the inverse problem of reconstructing the doping profile C from the VC map. In the sequel we shall consider two possible inverse problems for this map.

Current flow measurements through a contact

In this first inverse problem we assume that, for each C, the output is given by (0) U_j for some U_j. A realistic experiment corresponds to measure, for given {U_j}, with ||U_j|| small, the outputs

{(0)U_j | j = 1, ... , N}

(recall that S_C(0) = (V⁰,1,1)). In practice, the functions U_j are chosen to be piecewise constant on the contact G₁ and to vanish on G₀. From the definition of (0) we derive the following abstract formulation of the inverse doping profile problem for the VC map:

where

1) {U_j}Ì H^3/2(¶W_D) are fixed voltage profiles satisfying U_j|_G1 = 0;

2) Parameter: C = C(x) Î L²(W) = : X;

3) Output: Y = {(0) U_j}Î ^N = : ;

4) Parameter-to-output map: F: X ® .

The domain of definition of the operator F is

D(F) := {C Î L^¥ (W); C_m < C(x) < C_M, a.e. in W},

where C_m and C_M are suitable positive constants.

The inverse problem described above corresponds to the problem of identifying the doping profile C from the linearized stationary VC map at U = 0 (see bipolar case in Section 3).

The non-linear parameter-to-output operator F is well defined and Fréchet differentiable in its domain of definition D(F). This assertion follows from standard regularity results in PDE theory (see, e.g., [4], Propositions 2.2 and 2.3).

It is worth noticing that the solution of the Poisson equation can be computed a priori. The remaining problem (coupled system (9) for ( , )) is quite similar to the problem of EIT. In this inverse problem the aim is to identify the conductivity q = q(x) in the equation:

-div (qÑu) = f in W,

from measurements of the Dirichlet-to-Neumann map, which maps the applied voltage u|_¶W to the electrical flux q u_n|_¶W. The application (0) maps the Dirichlet data for and to the weighted sum of their Neumann data. It can be seen as the counterpart of electrical impedance tomography for common conducting materials.

Pointwise measurements of the current density

In the sequel, we investigate a different inverse problem related to the VC map. Differently from the previous paragraph, we shall assume that the VC operator maps the Dirichlet data for and in (9) to the sum of their Neumann data, i.e.

where functions V, , , J_n, J_p and U have the same meaning as in Section 3. It is immediate to observe that the Gateaux derivative of the VC map S_C at the point U = 0 in the direction h Î H^3/2(¶W_D) is given by

where ( , ) solve system (9). Notice that, for each applied voltage U, the VC map associates a scalar valued function defined on G₁. In this case, the outputs (0) U_j are in a data space which is larger than in the case of current flow measurements.

Again we can derive an abstract formulation of type (10) for the inverse doping profile problem for the linearized stationary VC map with pointwise measurements of the current density. The only difference to the framework described in the previous paragraph concerns the definition of the Hilbert space Y, which is now defined by:

3') Output: Y = {(0) U_j}Î L²(G₁)^N = : ;

The domain of definition of the operator F, remains unaltered.

It is immediate to observe that the model concerning current flow measurements carries less information about the unknown parameter than the model related to pointwise measurements does. In so far, the inverse problem related with the first measurement type is harder to solve.

5 A numerical experiment

In this section we apply numerical methods to solve an inverse doping profile problem related to the VC map. We consider the stationary linearized bipolar model with pointwise measurements of the current density.

In the sequel we consider the bipolar model introduced in Section 3. It follows from the assumption Q = 0 that the Poisson equation (6a) and the continuity equations (9a), (9b) decouple. The inverse doping profile problem corresponds to the identification of C = C(x) from pointwise measurements of the total current density J at the contact G₁, namely

(compare with the Gateaux derivative of the VC map S_C at the point U = 0 in (8)). Here (V⁰, , ) solve, for each applied voltage U, the system (6), (9), with h substituted by U.

Notice that we can split the inverse problem in two parts: First we define the function g(x) : = e^V0(x), x Î W , and solve the parameter identification problem

for g, from measurements of (µ_ng _n - µ_pg^-1

_n)|_G1. The second step consists in the determination of C in

C(x) = g(x) - g^-1(x) - l²D(In g(x)), x Î W.

Since the evaluation of C from g can be explicitely performed in a stable way, we shall focus on the problem of identifying the function parameter g in (12).

Summarizing, the inverse doping profile problem in the linearized bipolar model for pointwise measurements of the current density reduces to the identification of the parameter g in (12) from measurements of the DN map

If we take into account the restrictions imposed by the practical experiments described in Section 4, it follows:

i) The voltage profiles U Î H^3/2(¶W_D) must satisfy U |_G1 = 0;

ii) The identification of g has to be performed from a finite number of measurements, i.e. from the data

For this experiment concerning pointwise measurements of the current density, we assume that only one measurement is available, i.e. N = 1 in (13). What concerns the numerical implementation, we applied an iterative method of level set type to solve the identification problem for g in (12) (see [19, 15, 20]). The domain WÌ ² is the unit square, and the boundary parts are defined as follows

G₁ := {(x,1) ;x Î (0,1)}, G₀ := {(x, 0) ; x Î (0, 1)},

¶W_N := {(0, y) ; y Î (0, 1)} È {(1, y) ; y Î (0,1)}.

The fixed input U, is chosen to be a piecewise constant function supported in G₀

The doping profile to be reconstructed is shown in Figure 1(a). In Figure 1(b) the voltage source U (applied at G₀) and the corresponding solution of (12) are shown. In these pictures, as well as in the forthcoming ones, G₁ is the lower left edge and G₀ is the top right edge (the origin corresponds to the upper right corner).

In Figure 2 we present a numerical experiment for the bipolar model with pointwise measurements of the current density. Here exact data is used for the reconstruction of the p-n junction in Figure 1(a). The pictures show plots of the iteration error after 1, 10 and 100 steps of the level set method respectively.

What concerns the quality of the reconstruction of the P-N junction, the level set approach considered in this paper brings much better results than the Landweber-Kaczmarz approach implemented in [4]. A possible explanation for the different performance of these methods is the fact that the Landweber-Kaczmarz approach does not take into account the assumption that the coefficient g in (12) for such application is a piecewise constant function. The Landweber-Kaczmarz method tries to identify a real function defined on W, which is a much more complicated object than the original unknown curve (the P-N junction). Due to the nature of the level set approach, it incorporates in a natural way the assumption that g is piecewise constant in W.

Acknowledgments. The author was partially supported by the Brazilian National Research Council CNPq, grants 305823/03-5 and 478099/04-5.

Received: 01/IX/06. Accepted: 09/X/06.

#687/06.

Properties of silicon at room temperature

Relevant physical constants:

Permittivity in vacuum: e₀ = 8.85 ×10^-14 As V^-1 cm^-1;

Elementary charge: q = 1.6 ×10^-19 As.

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Appendix

1

Notice the applied potential has also to be rescaled:

U(

x) ¬

U(

x)/

U_T.

Publication Dates

• Publication in this collection
  19 Mar 2007
• Date of issue
  2006

History

• Received
  01 Sept 2006
• Accepted
  09 Oct 2006