JP3313281B2 - Method for evaluating characteristics of semiconductor device - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for evaluating characteristics of a semiconductor device, and more particularly to a method for evaluating the conductance of a MOS semiconductor device with high accuracy and high speed by numerical simulation.

[0002]

2. Description of the Related Art To calculate conductance by numerical simulation, it is necessary to calculate the distribution of potential ψ, electron density n, and hole density p inside a MOS type semiconductor device.

FIG. 6 is a diagram for explaining a conventional method for evaluating the characteristics of a semiconductor device. This diagram schematically shows the inside of the MOS type semiconductor device, and it is assumed that the portion is arranged in the order of metal, insulator, and semiconductor from the left in the drawing. In the conventional method described here, M
The OS type semiconductor device is divided into minute regions (hereinafter, referred to as control volumes) 1. Further, a point called a grid point X is arranged in each control volume 1. Then, ψ, n, and p on each grid point X are calculated. Ψ, n, p on each of these grid points X are interpreted as representing ψ, n, p in the corresponding control volume, and by connecting ψ, n, p in each control volume, the inside of the MOS type semiconductor device is The distribution of ，, n, p is calculated.

The conductance is determined by the product of the number of carriers and the carrier mobility (μ) in each region in the semiconductor. Therefore, it is necessary to calculate μ on each grid point. In a MOS semiconductor device, since the conductance near the interface between the insulator and the semiconductor (hereinafter, referred to as an insulator / semiconductor interface) of the MOS semiconductor device has a significant effect on the characteristics, the μ in this region can be accurately determined. It needs to be calculated.

In the prior art, μ is calculated from an empirical formula based on experimental values. However, in experiments, μ at each point of the semiconductor device cannot be measured, and only the average μ of the inversion layer formed near the insulator / semiconductor interface can be measured. Therefore, a method of calculating μ on each grid point based on an average μ obtained by experiment is required. In the prior art, this method uses the following s
The equation of hin is used.

[0006]

Μ (E) = μeff (Eeff) + (E−E0) d
μeff (Eeff) / dE (Equation 1) where μeff is the average μEeff = ηE + (1-η) E0 of the inversion layer (referred to as an effective electric field) E0 The electric field η = 1/2 outside the inversion layer Case) η = 1/3 (in the case of holes) N impurity concentration of semiconductor region ε dielectric constant of semiconductor k Boltzmann constant T temperature of semiconductor ni intrinsic carrier concentration of semiconductor φn pseudo-Fermi potential of electron φp pseudo-Fermi potential of hole It is.

Using the Shin's formula, μ on a grid point can be calculated as a function of the electric field E on the grid point. Here, all electric fields indicate components in a direction perpendicular to the insulator / semiconductor interface.

As described above, conductance near the insulator / semiconductor interface has a significant effect on characteristics. Μ in the control volume including the insulator / semiconductor interface is calculated from E on the corresponding grid point using the shin equation. In the prior art, this grid point is located at the insulator / semiconductor interface. Therefore, μ near the insulator / semiconductor interface is calculated from the electric field Eint at the insulator / semiconductor interface by using the equation of shin.

[0009]

However, in the conventional method of calculating the conductance by the method for evaluating the characteristics of a semiconductor device, as shown in FIG. 7, the obtained conductance depends on the setting of the control volume in the numerical simulation. There was a problem that the value of greatly changed. The cause will be described below.

FIG. 8 is a table showing the carrier mobility μ in the depth direction from the insulator / semiconductor interface. From this chart, it can be seen that the carrier mobility changes rapidly near the insulator / semiconductor interface. If the way of setting the control volume in the semiconductor area is changed due to this rapid change, there is a problem that the calculated conductance value changes.

[0011] This problem will be described in more detail. FIG.
FIG. 4 is a diagram for explaining a problem when the carrier mobility μ changes rapidly. In the figure, the vertical axis indicates the carrier mobility μ, and the horizontal axis indicates the distance from the insulator / semiconductor interface. Reference numeral 1 conceptually indicates the width of the control volume, and the graph in the figure is a simplified version of the graph showing the carrier mobility μ in FIG. As shown in FIG. 9A, when the control volume 1 is set large, the carrier mobility μ at a predetermined grid point of the control volume is set.
It can be seen that the use of int increases the error of the carrier mobility μ. In order to avoid this problem, in the related art, as shown in FIG. 9B, it is necessary to set a large number of extremely small control volumes near the insulator / semiconductor interface.

As a result, there has been a problem that a large amount of calculation time and computer memory are consumed in the numerical simulation. Further, when calculating the conductance using the practical calculation time and the memory of the computer, the calculation had to be performed with insufficient accuracy.

SUMMARY OF THE INVENTION The present invention has been made in view of the above circumstances, and an object of the present invention is to evaluate the characteristics of a semiconductor device in which the conductance of the semiconductor device can be accurately calculated with a short calculation time and computer resources. It is to provide a method.

[0014]

In order to achieve the above object, according to the first aspect of the present invention, a semiconductor device is divided into a minute region serving as a control volume, and conductance is calculated for the minute region by numerical simulation, and the characteristic is calculated. In the method of evaluating, the electric field at the interface between the insulator and the semiconductor of the semiconductor device, a vertical component of the interface electric field which is a component perpendicular to this interface is calculated, the control volume including the interface, and the control volume adjacent thereto An electric field at the boundary with the control volume on the semiconductor side to be calculated, a boundary electric field vertical component that is a component perpendicular to the interface is calculated, and an average of the interface electric field vertical component and the boundary electric field vertical component includes the interface. It is calculated as the electric field of the control volume, and the carrier mobility can be calculated using the calculated electric field. In characterized in that it comprises calculating a conductance.

In the configuration of the present invention, when calculating the electric field of the control volume, the average of the vertical component of the interface electric field and the vertical component of the boundary electric field is calculated as the electric field of the control volume including the interface, and the calculated electric field is used. The conductance is calculated by obtaining the carrier mobility by using the above method. By doing so, accurate conductance can be obtained even in the vicinity of the interface between the insulator and the semiconductor without the need to set the control volume small as in the conventional case. Therefore, the conductance can be calculated without depending on how to set the control volume in the numerical simulation. As a result, it is possible to avoid an increase in the calculation time for evaluating the characteristics of the semiconductor device and an increase in the use of hardware resources for the calculation.

According to a second aspect of the present invention, there is provided a method of dividing a semiconductor device into minute regions serving as control volumes, calculating conductance of the minute regions by numerical simulation, and evaluating characteristics thereof. Calculating the potential, electron density, and hole density inside the semiconductor device, and calculating the electric field at the interface between the insulator and the semiconductor of the semiconductor device from the calculated potential, electron density, and hole density. An interface electric field vertical component calculation step of calculating an interface electric field vertical component which is a component perpendicular to the interface; and an electric field at a boundary between a control volume including the interface and a semiconductor-side control volume adjacent to the control volume. A boundary for calculating a boundary electric field vertical component which is a component perpendicular to the interface. The average of the control volume using the electric field vertical component calculation step, the interface electric field vertical component obtained in the interface electric field vertical component calculation step, and the boundary electric field vertical component obtained in the boundary electric field vertical component calculation step. An electric field calculation step of a control volume for calculating an electric field, a mobility calculation step of a control volume for calculating a carrier mobility using the electric field obtained in the electric field calculation step of the control volume, and a mobility calculation of the control volume A conductance calculating step of calculating the conductance using the carrier mobility obtained in the step, and determining whether or not a predetermined convergence condition is satisfied; if not, returning to the potential etc. calculating step and calculating again A convergence determination step of performing Using the conductance is determined to have converged in step and performing characterization of the semiconductor device.

In the configuration of the present invention, when calculating the electric field of the control volume, the average of the vertical component of the interface electric field and the vertical component of the boundary electric field is calculated as the electric field of the control volume including the interface, and the calculated electric field is used. The conductance is calculated by calculating the carrier mobility in the same manner as in the first aspect of the present invention. Further, in the present invention, it is determined whether or not a predetermined convergence condition is satisfied, and if not, a recalculation is performed. Here, in determining whether the predetermined condition is satisfied, for example, a potential is obtained from the conductance calculated in the conductance calculation step,
The potential may be compared with the potential calculated in the potential etc. calculating step, and if the difference is within a predetermined range, the convergence condition may be satisfied. Similarly, an electron density and a hole density may be used, and a plurality of convergence conditions may be determined. Further, the conductance may be used.

Thus, according to the configuration of the present invention, the conductance can be calculated without depending on the setting of the control volume in the numerical simulation. As a result, it is possible to avoid an increase in the calculation time for evaluating the characteristics of the semiconductor device and the use of hardware resources for the calculation. Further, according to the present invention, the conductance can be evaluated with high accuracy.

As described above, in the method for evaluating the characteristics of a semiconductor device according to the present invention, the dependence of the obtained conductance value on the method of setting the control volume can be reduced. The case where the volume is smaller than the thickness of the inversion layer and the case where the volume is larger are based on different physical grounds, and therefore will be described in order.

First, the case where the size of the control volume including the insulator / semiconductor interface is smaller than the thickness of the inversion layer, that is, the case as shown in FIG. 3 will be described. The vertical axis in the figure indicates the insulator / semiconductor interface, and reference numeral 1 conceptually indicates the width of the control volume.

As described above, in the prior art, the insulator /
The μ of the control volume including the semiconductor interface is given as μ (μint) on the insulator / semiconductor interface.
In this method, the condition under which conductance can be accurately calculated is that the average μ (μave) in the control volume including the insulator / semiconductor interface is sufficiently close to μint. To achieve this, the control volume including the insulator / semiconductor interface has to be made extremely small. This is because μ changes very steeply near the insulator / semiconductor interface as shown in FIG. Further, an error in conductance calculated without making the control volume extremely small is proportional to the size of the control volume. This is the reason that the conductance calculated by the conventional technique strongly depends on the size of the control volume.

In order to calculate conductance accurately with a larger control volume, μave of the control volume including the insulator / semiconductor interface may be given as μ of the control volume. As shown in FIG. 3, in the vicinity of the insulator / semiconductor interface, μ has a linear relationship with the distance from the insulator / semiconductor interface. Therefore, the following relationship is established.

Μave = (μint + μbs) / 2 (Equation 2) Here, μbs is μ at a boundary between a control volume including an insulator / semiconductor interface and a semiconductor-side control volume adjacent thereto.

Further, based on Shin's equation, μ is tailored by E and approximated by ignoring the second and subsequent terms,

Μave = (μ (Eint + μ (Ebs)) / 2−μ ((Eint + Ebs) / 2) = μ (E ′) (Equation 3)

That is, in the method for evaluating the characteristics of a semiconductor device according to the present invention, μave of the control volume including the insulator / semiconductor interface is calculated by substituting E ′ into the equation of shin to obtain μ. Therefore, an accurate conductance can be calculated without depending on the size of the control volume.

Next, the case where the size of the control volume including the insulator / semiconductor interface is larger than the thickness of the inversion layer will be described. In this case, the boundary between the control volume including the insulator / semiconductor interface and the adjacent control volume on the semiconductor side is outside the inversion layer. Therefore, Ebs = E0 (Equation 4). With this in mind,

Μ (E ′) = μeff (ηEint + (1−η) E0− (η / 2) (Eint−E0)) + (η / 2) (Eint−E0) dμeff (ηEint + (1−η) E0− (η / 2) (Eint−E0)) / dEeff (Equation 5) When μeff is Taylor-expanded by Eeff and an approximation that ignores the second and subsequent terms is given by μ (E ′) = Μeff (Eeff) (Equation 6) where Eeff = ηEint + (1-η) E0. That is, the average μ of the inversion layer is given as μ of the control volume including the insulator / semiconductor interface.

FIG. 4 (b) shows the carrier concentration near the insulator / semiconductor interface. The hatched portion in the figure indicates the carrier concentration in the depth direction. As described above, carriers accumulate in a region near the semiconductor at the insulator / semiconductor interface (hereinafter, referred to as an inversion layer) and hardly exist in other regions. It turns out that it is not necessary. Since the control volume 1 includes the inversion layer, all carriers forming the inversion layer are included in the control volume. Therefore, it is sufficient to give the average μ of the inversion layer. Accordingly, the method for evaluating characteristics of a semiconductor device according to the present invention can calculate an accurate conductance even when the size of the control volume including the insulator / semiconductor interface is larger than the thickness of the inversion layer. You can.

As described above, according to the method for evaluating the characteristics of a semiconductor device according to the present invention, the conductance can be calculated with high accuracy and high speed without depending on the method of setting the control volume.

[0029]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, embodiments of a method for evaluating characteristics of a semiconductor device according to the present invention will be described with reference to the drawings.

FIG. 1 is a flowchart showing an embodiment of a method for evaluating characteristics of a semiconductor device according to the present invention. The method for evaluating the characteristics of a semiconductor device according to the present invention includes a step S101 for calculating a potential, an electron density, and a hole density in a semiconductor device, and a step S101 for calculating a potential, an electron density, and a hole density. An interface electric field Eint for calculating an electric field Eint at an interface between an insulator and a semiconductor of the semiconductor device.
A calculating step S102, and a boundary electric field Ebs calculating step S103 for calculating an electric field Ebs at an interface between the control volume including the interface and the semiconductor-side control volume adjacent to the control volume.
And an interface electric field Eint obtained in the interface electric field Eint calculation step S103 and the boundary electric field Ebs obtained in the boundary electric field Ebs calculation step S104, for calculating the average electric field of the control volume. Electric field E ′ calculation step S104 and control volume mobility μ calculation step S105 of calculating carrier mobility using the electric field obtained in the control volume electric field calculation step S104.
And a conductance calculating step S1 of calculating a conductance using the carrier mobility obtained in the control volume mobility μ calculating step S105.
06, and a convergence determination step of determining whether or not a predetermined convergence condition is satisfied. If the predetermined convergence condition is not satisfied, returning to the potential etc. calculation step and performing calculation again is included. The characteristic of the semiconductor device is evaluated using the conductance determined as follows.

Next, the operation of the method for evaluating characteristics of a semiconductor device according to the present embodiment will be described with reference to the drawings. In implementing the present embodiment, it is assumed that a semiconductor device for which characteristics are to be evaluated in advance is divided into control volumes 1 using a conventional technique or the like. Characteristic evaluation is performed on each of the divided control volumes. FIG. 2 is a diagram for explaining the present embodiment. As shown in this figure, the inside of the MOS type semiconductor device is schematically shown, and a metal, an insulator, and a semiconductor are arranged in this order from the left in the drawing. A point called a lattice point X is arranged inside. The potential ψ at the grid point X of each control volume 1 is assumed to be the potential at the control volume.

First, in a potential etc. calculating step S101,
Ψ, n, p on each grid point are calculated. Using this value,
The electric field at the interface between the insulator and the semiconductor (interface electric field Eint) is calculated (step S102). Interface electric field E calculated here
int is

Eint = − (算出 (i + 1) −ψ (i)) / (x (i + 1) − (x (i)) / ε (Equation 7) In the boundary electric field Ebs calculation step S103, the control volume including the interface
An electric field (boundary electric field Ebs) at the boundary with the control volume on the semiconductor side adjacent thereto is calculated. The boundary electric field Ebs used here is

Calculated using the following equation: Ebs = − (ψ (i) −ψ (i−1)) / (x (i) − (x (i−1)) / ε (Equation 8)

Subsequently, in step S104 for calculating the electric field E ′ of the control volume, when calculating μ in the control volume including the insulator / semiconductor interface, first, E ′ is calculated by the following equation.

E ′ = (Eint + Ebs) / 2 (Equation 9) Here, Eint and Ebs are respectively an interface electric field Eint calculation step s102 and a boundary electric field Ebs calculation step S1.
The one calculated in 03 is used.

Subsequently, in a control volume mobility μ calculation step S105, μ is calculated by substituting the obtained electric field E ′ of the control volume into the equation of shin.

Μ = μ (E ′) (Equation 10) Since (Equation 10) is physically equivalent to the following equation, μ may be calculated using the following equation.

[0037]

Μ = {(Eint−E0) μeff (ηEint + (1−η) E0) − (Ebs−E0) μeff (ηEbs + (1−η) E0)} / (Eint−Ebs) (Equation 11) Subsequently, in the conductance calculation step S106, the conductance in the control volume is calculated from the calculated mobility μ of the control volume. This calculation method can be performed using a conventional technique.

Subsequently, in the convergence determination step S107,
The potential may be obtained from the obtained conductance, and may be compared with the potential calculated in the potential etc. calculating step S101. If the difference is within a predetermined range, the convergence condition may be satisfied. Similarly, an electron density and a hole density may be used, and a plurality of convergence conditions may be determined. Further, the conductance may be used. As a result of this comparison, if the difference is out of the predetermined range, a potential or the like is calculated from the conductance in a potential or the like calculation step S101. On the other hand, when the predetermined convergence condition is satisfied, the conductance obtained in the conductance calculation step S106 is set as the conductance of the control volume, and the characteristic of the semiconductor device is evaluated using the conductance.

FIG. 5 is a table showing the conductance calculated by using the method for evaluating the characteristics of the semiconductor device according to the present embodiment. This figure shows the case where the control volumes used for the numerical simulation are 1Å, 1 nm, and 10 nm. As can be seen from this figure, almost equal regardless of the size of the control volume, that is,
It can be seen that the conductance is calculated without depending on the size of the control volume.

[0040]

As described above, the method for calculating the conductance of a semiconductor device according to the present invention can calculate the conductance without depending on the method of setting the control volume in the numerical simulation.

For this reason, a large control volume can be set in the vicinity of the insulator / semiconductor interface, so that accurate conductance can be calculated using a short calculation time and a computer memory.

[Brief description of the drawings]

FIG. 1 is a flowchart illustrating a procedure for calculating μ in a control volume including an insulator / semiconductor interface by a method for evaluating characteristics of a semiconductor device according to an embodiment.

FIG. 2 is a schematic diagram illustrating a control volume used in the embodiment.

FIG. 3 is a diagram for explaining a method for obtaining an average mobility according to the embodiment.

FIG. 4 is a diagram for explaining a method of obtaining an average mobility according to the embodiment.

FIG. 5 is a characteristic diagram of the conductance of the MOS semiconductor device calculated according to the present embodiment.

FIG. 6 is a schematic diagram showing a method for evaluating characteristics of a semiconductor device according to a conventional method.

FIG. 7 is a characteristic diagram of the conductance of a MOS semiconductor device calculated by a conventional technique.

FIG. 8 is a diagram showing the depth dependence of μ of a MOS semiconductor device.

FIG. 9 is a diagram for explaining a method of obtaining carrier mobility by a conventional method.

[Explanation of symbols]

 1 control volume

──────────────────────────────────────────────────続 き Continued on the front page (51) Int.Cl. ⁷ Identification code FI H01L 29/78

Claims (2)

(57) [Claims] 1. A method of dividing a semiconductor device into minute regions serving as control volumes, calculating conductance of the minute regions by numerical simulation, and evaluating characteristics of the minute regions. Calculating an interface electric field vertical component that is a component perpendicular to the interface, the electric field at the interface, and the electric field at the boundary between the control volume including the interface and the control volume on the semiconductor side adjacent to the control volume; A boundary electric field vertical component which is a component perpendicular to the boundary electric field is calculated, an average of the interface electric field vertical component and the boundary electric field vertical component is calculated as an electric field of a control volume including the interface, and a carrier is calculated using the calculated electric field. Calculating conductance by determining mobility. Characteristic evaluation method of a semiconductor device. 2. A method of dividing a semiconductor device into minute regions serving as control volumes, calculating conductance of the minute regions by numerical simulation, and evaluating characteristics of the minute regions. A potential calculating step for calculating a hole density; and an electric field at an interface between the insulator and the semiconductor of the semiconductor device from the calculated potential, the electron density, and the hole density. An interface electric field vertical component calculating step of calculating an interface electric field vertical component, which is a component; an electric field at a boundary between a control volume including the interface and a semiconductor-side control volume adjacent to the control volume, the electric field being perpendicular to the interface Electric field vertical component calculation step for calculating the boundary electric field vertical component And calculating the average electric field of the control volume using the interface electric field vertical component obtained in the interface electric field vertical component calculation step and the boundary electric field vertical component obtained in the boundary electric field vertical component calculation step. A control volume electric field calculation step, a control volume mobility calculation step of calculating carrier mobility using the electric field obtained in the control volume electric field calculation step, and a control volume mobility calculation step. A conductance calculation step of calculating conductance using the obtained carrier mobility; and determining whether a predetermined convergence condition is satisfied. If not, returning to the potential etc. calculation step and performing calculation again And a convergence determination step. Characterization method of a semiconductor device which is characterized in that the characterization of the semiconductor device using the determined conductance that.