JP3692005B2 - Semiconductor element evaluation apparatus, semiconductor element evaluation method, and computer-readable recording medium recording semiconductor element evaluation program - Google Patents

Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a simulator which is a semiconductor element evaluation apparatus for evaluating electrical characteristics of a MIS (Metal Insulator Semiconductor) type field effect transistor.
[0002]
[Prior art]
In recent years, device structures have become more complicated with the miniaturization and higher functionality of LSIs. Therefore, in order to develop an optimum structure as a product, relying only on prototyping leads to a long development period and an increase in cost. Therefore, a simulator, which is a semiconductor element evaluation apparatus, plays a major role as means for reducing the number of prototypes and providing an optimally structured solution. In particular, the simulation technique for the MIS transistor, which is the central device of LSI, is one of the most required accuracy and calculation speed.
[0003]
When a MIS field effect transistor is miniaturized, a short channel effect is generated in which a threshold value of a gate voltage (hereinafter referred to as a threshold voltage) is lowered. The short channel effect is a phenomenon in which the threshold voltage sharply decreases as the channel length of a transistor decreases. When the threshold voltage is lowered, a leakage current flowing through the transistor is increased even in the off state, and standby power of a portable information device or the like is increased. In addition, a large change in the threshold voltage with respect to the channel length causes a slight fluctuation in the channel length at the time of manufacturing leads to a change in the threshold voltage, leading to a deterioration in the final LSI yield. The short channel effect is generated when the depletion layer in the drain region spreads in the channel region or the substrate region due to the shortening of the channel length. In order to suppress the spread of the depletion layer, it is necessary to increase the substrate impurity concentration and the channel impurity concentration.
[0004]
However, if these impurity concentrations are increased to sufficiently suppress the short channel effect, the threshold voltage may become too high. This reduces the voltage difference between the on state and the off state, making stable circuit operation difficult. In order to suppress this increase in threshold voltage, it is effective to reduce the thickness of the gate insulating film. As a result, the capacitance between the gate electrode and the channel region (gate capacitance) increases, so that the voltage at which carriers start to flow in the channel region decreases, and the increase in threshold voltage due to the increase in substrate concentration is offset. In this way, the threshold voltage is adjusted to be about 20% of the power supply voltage. For example, if the power supply voltage is 1.5V, the threshold voltage is 0.3V.
[0005]
The increase in the substrate impurity concentration and channel impurity concentration and the thinning of the gate insulating film described above increase the vertical electric field in the gate electrode direction at the interface of the channel region in contact with the gate insulating film. This reveals the effect of quantization at the interface. That is, the carrier wave function is localized at the interface, and the maximum value of the carrier concentration distribution is shifted to a position about 10 mm away from the interface. Therefore, since the gate insulating film is effectively thickened, the threshold voltage increases. Incorporating this effect into device simulation is essential to ensure the accuracy of simulation technology.
[0006]
In order to calculate the quantum effect accurately, it is necessary to solve the Schrodinger equation (for example, F. Stern and WE Howard, "Properties of semiconductor surface inversion layers in the electric quantum limit," Phys. Rev., vol. 163 , p. 816, 1967.). However, this method is effective in a simple one-dimensional structure (for example, a direction extending from the gate electrode through the gate insulating film to the channel region), and cannot be applied to an actual device.
[0007]
Known as a means of effectively incorporating quantum effects into device simulations is the use of quantum correction potentials (e.g. MG Ancona, Z. Yu, W.-C. Lee, RW Dutton, and PV Voorde, "Density-gradient simulation of quantum effects in ultra-thin-oxide MOS structures," p. 97, SISPAD '97.). For example, the description of electrons is as follows.
[0008]
qnv _e = μ [qn∇ (ψ + Δψ _q ) -k _B ∇nT] (1)
Δψ _q = β∇ ・ ∇√n / √n (2)
β = h ² / (24π ² qm ^* ) (3)
Where (1) is the current density equation, q is the unit elementary charge, n is the electron concentration, v _e is the electron velocity vector, μ is the electron mobility, ∇ is the spatial differential vector, ψ is the potential, and Δψ _q is Quantum correction potential, k _B is Boltzmann constant, T is electron temperature, h is Planck's constant, and m ^* is effective mass. For convenience of explanation, the method using the equations (1) to (3) is called an ancona method.
[0009]
This Ancona method is an attempt to effectively introduce a quantum effect by generating a potential for reducing the electron concentration at the interface between the gate insulating film and the channel region by Δψ _q in the current density equation of equation (1). However, it is known that it is difficult to obtain a convergent solution by this method. That is, in the device simulation, the Poisson equation and the current continuity equation are solved in a self-consistent manner by iterative calculation to obtain the distribution of the electron concentration n and the potential ψ. The current density formula (1) is used as an auxiliary formula for the current continuity formula. In this case, equation (2) is also used in the iterative calculation, but the electron concentration n in the equation is a value that is in the middle of convergence. The carrier concentration distribution in the middle of convergence is often unstable, and the differential value becomes very unstable. For this reason, an unnatural Δψ _q is calculated, and iterative calculations are divergent and a solution cannot often be obtained.
[0010]
Furthermore, after obtaining a convergence solution by classical device simulation that does not consider the quantum effect, the result can be used as the initial value of the Ancona method described above (for example, ATLAS User's Manual, vol. 1). , p. 3-88, Nov, 1998.). However, in that case multiplied by a damping coefficient in front of [Delta] [phi] _q, the damping coefficient toward zero to 1 is gradually changed. This method takes 5 to 10 times more computation time than the classical simulation and is not necessarily guaranteed to converge. The classical simulation is to solve equation (1) with Δψ _q = 0.
[0011]
[Problems to be solved by the invention]
An object of the present invention is to improve poor convergence and slow calculation speed when a quantum effect is introduced into a device simulator which is a semiconductor element evaluation apparatus.
[0013]
The present invention is a semiconductor device evaluation apparatus for evaluating an insulated gate field effect transistor,
Means for solving Poisson's equation which is a relational expression of potential, electron concentration and hole concentration;
Using the potential, electron concentration, and hole concentration temporarily obtained by the means for solving the Poisson equation, the potential is ψ, the unit elementary charge is q, h ² / (24π ² qm ^* ) (where h is Planck's constant, Quantum correction represented by the relational expression Δψ _q = β∇ · ∇√n / √n, where β is the quantum correction potential coefficient represented by m ^* is effective mass, q is unit elementary charge) , and n is the electron concentration. Means for calculating the potential Δψ _q ;
Using the quantum correction potential Δψ _q temporarily obtained by the means for calculating the quantum correction potential Δψ _q , the current is an auxiliary equation of the Poisson equation, the current continuity formula of electron concentration and hole concentration, and the current continuity formula Having a means for solving the density equation and obtaining self-consistent potential, electron concentration and hole concentration,
N in the relational expression of the quantum correction potential Δψ _q is a function satisfying the Schrödinger equation , the unit elementary charge q, the effective mass m ^* , the dielectric constant ε, the Planck constant h, and the depletion layer formed in the semiconductor substrate. B = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32) N], which is a function of the surface density N _depl of the impurity charge and the surface density N _inv of the electrons inverted in the semiconductor substrate. _inv ]] ^1/3 , the function satisfying the Schrödinger equation is expressed as Az ² exp (−bz) where A is the proportionality coefficient and z is the distance from the surface of the semiconductor substrate to the depth direction. It represented, the quantum correction potential is a semiconductor device evaluation apparatus characterized that you denoted _{Δψ q (z) = βb (} b / 4-1 / z).
[0015]
The present invention is a semiconductor element evaluation method for evaluating an insulated gate field effect transistor,
Solve Poisson's equation, which is the relational expression of potential, electron concentration and hole concentration,
By solving the Poisson equation, the potential is ψ, the unit charge is q, h ² / (24π ² qm ^* ) (where h is Planck's constant ) using the temporarily obtained potential, electron concentration and hole concentration. , M ^* is an effective mass, q is a unit elementary charge) , and a quantum correction potential coefficient represented by β and an electron concentration n is represented by a relational expression of Δψ _q = β∇ · ∇√n / √n. Quantum correction potential Δψ _q
Using the quantum correction potential Δψ _q temporarily obtained by calculating the quantum correction potential Δψ _q , the current is an auxiliary equation of the Poisson equation, the current continuity formula of electron concentration and hole concentration, and the current continuity formula A method for solving a density equation and obtaining self-consistent potential, electron concentration and hole concentration,
N in relations of the quantum correction potential [Delta] [phi] _q is Ri functions der satisfying Schrodinger equation, the unit elementary charge q, effective mass m ^*, the dielectric constant epsilon, the depletion layer formed Planck constant h, the semiconductor substrate B = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32)], which is a function of the surface density N _depl of the impurity charges and the surface density N _inv of the electrons inverted in the semiconductor substrate. N _inv ]] ^1/3 , the function satisfying the Schrödinger equation is represented by Az ² exp (−bz) where A is the proportionality coefficient and z is the distance from the surface of the semiconductor substrate in the depth direction. And the quantum correction potential is expressed as Δψ _q (z) = βb (b / 4−1 / z) .
[0016]
The present invention is a computer-readable recording medium recording a semiconductor element evaluation program for evaluating an insulated gate field effect transistor,
A procedure for solving Poisson's equation which is a relational expression of potential, electron concentration and hole concentration;
Using the potential, electron concentration and hole concentration temporarily obtained by the procedure for solving the Poisson equation, the potential is ψ, the unit elementary charge is q, h ² / (24π ² qm ^* ) (where h is Planck's constant, Quantum correction represented by the relational expression Δψ _q = β∇ · ∇√n / √n, where β is the quantum correction potential coefficient represented by m ^* is effective mass, q is unit elementary charge) , and n is the electron concentration. A procedure for calculating the potential Δψ _q ;
Using the quantum correction potential Δψ _q temporarily obtained by the procedure of calculating the quantum correction potential Δψ _q , the current is an auxiliary equation of the Poisson equation, the current continuity equation of electron concentration and hole concentration, and the current continuity equation Having a procedure for solving the density equation and obtaining self-consistent potential, electron concentration and hole concentration,
N in relations of the quantum correction potential [Delta] [phi] _q is Ri functions der satisfying Schrodinger equation,
A function of unit elementary charge q, effective mass m ^* , dielectric constant ε, Planck's constant h, surface density N _depl of impurity charges in the depletion layer formed in the semiconductor substrate, and surface density N _inv of electrons inverted in the semiconductor substrate When b is expressed as b = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32) N _inv ]] ^1/3 , a function satisfying the Schrodinger equation has a proportional coefficient: A, where z is the distance in the depth direction from the surface of the semiconductor substrate, it is expressed as Az ² exp (−bz), and the quantum correction potential is Δψ _q (z) = βb (b / 4−1 / z) is a computer-readable recording medium on which a semiconductor element evaluation program is recorded.
[0017]
According to the present invention, since the quantum correction potential is introduced as a simple distance function from the interface between the gate insulating film and the semiconductor substrate, the calculation is easy. In addition, when calculating the quantum correction potential, instead of calculating the differential value of the local carrier concentration distribution as in the past, the integral value from the interface is used, so the calculation is performed due to local carrier concentration fluctuations. There is no instability. Due to these effects, it is possible to perform calculation with good convergence and high speed.
[0018]
DETAILED DESCRIPTION OF THE INVENTION
FIG. 1 is a graph showing the spatial distribution of the quantum correction potential obtained by the embodiment of the present invention. This is a distribution in the silicon substrate 1 in the AA ′ depth direction at the center of the channel of the electron conduction type insulated gate field effect transistor schematically shown in FIG. Since it is an electron conduction type, electrons are induced in the channel region. When a negative quantum correction potential is generated at the channel surface portion as shown in FIG. 1, the induced electrons are negatively charged, and thus are moved away from the channel surface. This effectively introduces the effect of reducing the wave function, which is the existence probability of electrons, on the channel surface.
[0019]
Hereinafter, how to obtain the spatial distribution of the quantum correction potential shown in FIG. 1 will be described.
[0020]
The impurity concentration of the silicon substrate 1 is 1 × 10 ¹⁸ cm ⁻³ of acceptor type (p type), the thickness of the gate oxide film 2 is 5 nm, the gate electrode 3 is polycrystalline silicon, and the impurity concentration is donor type (n type). 1 × 10 ²⁰ cm ^-3, the source n ⁺ impurity concentration of the substrate surface with a donor-type diffusion layer 4 and the drain n ⁺ diffusion layer 5 and the depth at 1 × 10 ²⁰ cm ^-3 is is 50nm.
[0021]
The source electrode 6 and the drain electrode 7 are aluminum, and the protective insulating films 8 and 9 are silicon oxide films. The bias condition is that the gate voltage is V _g = 3 V and the drain voltage is V _d = 50 mV. Quantum correction potential [Delta] [phi] _q shown in FIG. 1, (2) by substituting a function that satisfies the Schrodinger equation electron concentration n is trial function (4) in the formula, a simple [Delta] [phi] _q (z) for ( 6) Equation is obtained.
[0022]
n (z) = Az ² exp (-bz) (4)
b = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32) N _inv )] ^1/3 (5)
Δψ _q (z) = βb (b / 4-1 / z) (6)
Where A is the proportional coefficient, z is the distance from the surface of the silicon substrate 1 in the depth direction, ε is the dielectric constant, N _depl is the surface density of the impurity charge in the depletion layer near the surface, and N _inv is the inverted electron Is the surface density. As for the hole concentration p, the quantum correction potential is calculated by the function of p (z) in the same manner from the equation (4) using the effective mass of holes. Here, “area density” refers to the number of carriers per unit area. The term “concentration” simply refers to the number of carriers per unit volume.
[0023]
FIG. 3 shows an algorithm of the calculation execution unit according to the embodiment of the present invention. First, in step 1 (S1), initial values of potential and carrier concentration distribution are set based on the impurity concentration and the previous simulation result. In other words, when the calculation execution unit is called for the first time, the initial values of the electron concentration n and the hole concentration p are obtained by simultaneously solving the charge neutrality condition and the constant np product condition of the following equation.
[0024]
p-n + N _d -N _a = 0 (charge neutral condition) (7)
np = n _i ² (Constant np product condition) (8)
Here, N _d is a donor-type impurity concentration, N _a is an acceptor-type impurity concentration, and n _i is an intrinsic carrier concentration. In the case of Si, n _i = 1 × 10 ¹⁰ cm ⁻³ . Regarding the potential ψ, for example, a semiconductor region of the same type that continues to a certain electrode is made equal to the potential of that electrode. That is, the potential in the source n ⁺ diffusion layer 4 is the same potential as the source electrode 6, the potential in the drain n ⁺ diffusion layer 5 is the same potential as the drain electrode 7, and the potential in the p-type silicon substrate 1 is the substrate electrode (the main electrode). In the embodiment, it is grounded). The electron concentration, hole concentration, and potential in the oxide film are zero. When the calculation execution unit is called for the second time, the previous calculation result is used as an initial value, and when called for the third time or later, the value extrapolated based on the previous and previous calculation results is initialized. Use as a value.
[0025]
Next, in step 2 (S2), only the Poisson equation described later is solved. At this time, the carrier concentration distribution may be estimated from the potential distribution assuming an energy distribution function such as Boltzmann statistics. Further, the calculation of the Poisson equation in S2 may be repeated until the potential distribution converges to some extent. As the energy distribution function, for example, assuming Boltzmann statistics, the electron concentration n and the hole concentration p are updated by the following equations.
[0026]
n = n ₀ exp (δψ / (k _B T _e )) (9)
p = p ₀ exp (-δψ / (k _B T _h )) (10)
Here, n ₀ and p ₀ is the electron concentration and hole concentration before correction, [Delta] [phi] is the correction amount of the potential, the T _e is the electron temperature, T _h is the hole temperature.
[0027]
Next, in step 3 (S3), a quantum correction potential that is a quantum correction potential is calculated. At this time, N _depl and N _inv are obtained by integrating the net impurity concentration distribution (N _d −N _a ) and the carrier concentration distribution in the substrate depth direction from the calculation point of each interface. The integration range may be a depletion layer width or a predetermined value may be input. The depletion layer width may be a point at which a change in integration result becomes a predetermined value, for example, 1% or less when N _inv is obtained from the interface by numerical integration.
[0028]
Next, in step 4 (S4), the following Poisson equation (11) and current continuity equations (12) and (13) are solved in a self-consistent manner.
[0029]
∇ ・ ε∇ψ = -q (p-n + N _d -N _a ) (11)
∂n / ∂t + ∇ ・ nv _e = GR (12)
∂p / ∂t + ∇ ・ pv _h = GR (13)
Here, N _d is the donor impurity concentration, N _a is the acceptor impurity concentration, GR is the generation and annihilation term carrier. (12) nv _e in formula mentioned above (1) _{(qnv e = μ [qn∇ (} ψ + Δψ q) -k B ∇nT]) using equation of current density including quantum correction potential of. The formula pv _h of the current density of holes is calculated by reversing the sign of the first term on the right side of the equation (1) and changing the model of mobility μ for holes. Note that v _h is a hole velocity vector. As a self-consistent solving method, for example, a partial differential equation is discretized by a finite difference method, converted into an algebraic equation, and solved using a Newton method (for example, edited by Dan Ryo, `` Process, Device, Simulation technology ", Section 3.3, Industrial books, published in 1988). Next, convergence determination is performed in step 5 (S5), and if not sufficiently converged, the process returns to S3. Here, the convergence determination uses at least one correction amount of potential, electron concentration, and hole concentration. For example, when at least one of δψ <1 × 10 ⁻³ , δn / n <0.01, and δp / p <0.01 is satisfied, it is determined that convergence has occurred. Here, Δn and Δp are correction amounts of the electron and hole concentrations.
[0030]
FIG. 4 shows the overall algorithm of the device simulation system according to the present invention. First, in module 1 (M1), a simulation structure, impurity concentration distribution, and bias conditions are input. Next, in module 2 (M2), the time (transient analysis time) and the voltage (gate, source, drain and substrate voltage) of each terminal of the device are set. Next, the simulation described in FIG. 3 is executed in the module 3 (M3). Next, in module 4 (M4), it is determined whether or not all the calculation conditions have been completed. If not completed, the process returns to M2, and the next time and the voltage of each terminal are set. When completed, the module 5 (M5) outputs the calculation result.
[0031]
The program for realizing the evaluation of the semiconductor element described above can be stored in a computer-readable recording medium. By reading this recording medium into a computer system, executing the program, and controlling the computer according to a predetermined processing procedure, the above-described semiconductor element evaluation can be realized. Here, examples of the recording medium include devices capable of recording a program such as a memory device, a magnetic disk device, an optical disk device, and a magnetic tape device.
[0032]
FIG. 5 is an external view showing an example of a computer system that reads a program stored in these recording media and realizes a semiconductor element evaluation process according to a procedure described therein. A floppy disk drive 81 and a CDROM drive 82 are provided on the front surface of the main body of the computer system 80, and a floppy disk 83 as a magnetic disk device or a CD-ROM 84 as an optical disk device is inserted from each drive inlet, By performing a predetermined read operation, the programs stored in these storage media can be installed in the system. Further, by connecting a predetermined drive device, for example, a ROM 85 as a memory device used in a game pack or the like, or a cassette tape 86 as a magnetic tape device can be used.
[0033]
FIG. 6 is a graph showing the electron concentration distribution in the depth direction calculated using the embodiment of the present invention. A dotted line (Classical) is a device simulation result when quantum correction is not included. The electron concentration is maximized at the interface. This is a result of solving the Schrodinger equation in a self-consistent manner with the Poisson equation, assuming that the broken line (Schrodinger) has a simple one-dimensional structure (direction AA ′ in FIG. 2). At the interface, the electron concentration decreases due to the quantum effect.
[0034]
Here, assuming a simple one-dimensional structure with a broken line (Schrodinger) (direction AA ′ in FIG. 2), a method of solving the Schrodinger equation with the Poisson equation in a self-consistent manner will be described.
[0035]
In the inversion layer of the MOSFET, the energy level interval spreads due to the quantum mechanical size effect, and these widened levels are called subbands. The Schrodinger equation for the i-th subband is given by
[0036]
P _i '' (z) = (8πm _i ^* / h ² ) [ψ (z) -E _i ] P _i (z) (14)
Here, P _i (z) is a wave function, P _i ″ (z) is a second-order spatial derivative in the z direction, E _i is an eigenvalue, and m _i ^* is an effective mass. P _i (z) satisfies the following equation.
[0037]
∫P _i (z) dz = 1 (integral range from zero to ∞) (15)
n (z) = Σ _i N _i P _i ² (z) (16)
N _i = a _i ln [1 + exp [(E _f -E _i ) / k _B T]] (17)
a _i = 4πn _vi m _i ^* k _B T / h ² (18)
Where E _f is Fermi energy and n _vi is a degenerate number. ψ (z) is updated by substituting n (z) into the Poisson equation.
[0038]
On the other hand, the solid line is the electron concentration distribution obtained by the embodiment of the present invention. Similar to the Schrodinger equation, the result reflects the quantum effect. In the present invention, as a matter of course, a simulation including the structure of all the semiconductor elements in FIG. 2 is performed.
[0039]
V _d is the drain voltage, V _g is the gate voltage, W is the gate width (depth direction in FIG. 6), L is the gate length, t _ox is the thickness of the gate insulating film, and X _j is the source / drain from the substrate surface. The depth of the region, N _sub, is the surface density of the net impurity in the channel impurity or p-type silicon substrate.
[0040]
FIG. 7 shows the current-voltage characteristics obtained by the embodiment of the present invention. A broken line (Classical) is a result when the quantum correction potential is not included, and a solid line is a result when the quantum correction potential of the present embodiment is included. It is confirmed that the higher the substrate impurity concentration is, the higher the threshold voltage is due to the quantum effect, and it is possible to simulate the shift of the IV characteristic to the higher gate voltage. Note that the calculation time of the present embodiment is only about twice as long as that when the quantum correction potential is not included, and is about 1 minute with the CPU 300 MHz and the memory 256 MB.
[0041]
According to the present invention, it is possible to perform virtual prototyping with high accuracy when designing a sub 0.1 μm generation device, which contributes to the reduction in the number of prototyping, the development period, and the LSI development cost.
[0042]
【The invention's effect】
ADVANTAGE OF THE INVENTION According to this invention, when introducing a quantum effect into the device simulator which is a semiconductor element evaluation apparatus, bad convergence and slow calculation speed can be improved.
[Brief description of the drawings]
FIG. 1 is a graph showing a spatial distribution of quantum correction potentials obtained by an embodiment of the present invention.
FIG. 2 is a schematic cross-sectional view of an electron conduction type insulated gate field effect transistor.
FIG. 3 is an algorithm of a calculation execution unit according to the embodiment of the present invention.
FIG. 4 is an overall algorithm of a device simulation system according to an embodiment of the present invention.
FIG. 5 is an external view showing an example of a computer system that reads a program stored in a recording medium and realizes data compression / decompression processing according to a procedure described therein.
FIG. 6 is a graph showing an electron concentration distribution in the depth direction calculated using the embodiment of the present invention.
FIG. 7 is a graph showing current-voltage characteristics obtained by the embodiment of the present invention.
[Explanation of symbols]
1 p-type silicon substrate 2 gate oxide film 3 n-type polycrystalline silicon gate electrode 4 source n ⁺ diffusion layer 5 drain n ⁺ diffusion layer 6 source electrode 7 drain electrodes 8 and 9 protective silicon oxide film 80 computer system 81 floppy disk Drive 82 CDROM drive 83 Floppy disk 84 as magnetic disk device CD one ROM as optical disk device
85 ROM as a memory device
86 Cassette tape as a magnetic tape unit

Claims (3)

A semiconductor device evaluation apparatus for evaluating an insulated gate field effect transistor,
Means for solving Poisson's equation which is a relational expression of potential, electron concentration and hole concentration;
Using the potential, electron concentration, and hole concentration temporarily obtained by the means for solving the Poisson equation, the potential is ψ, the unit elementary charge is q, h ² / (24π ² qm ^* ) (where h is Planck's constant, Quantum correction represented by the relational expression Δψ _q = β∇ · ∇√n / √n, where β is the quantum correction potential coefficient represented by m ^* is effective mass, q is unit elementary charge) , and n is the electron concentration. Means for calculating the potential Δψ _q ;
Using the quantum correction potential Δψ _q temporarily obtained by the means for calculating the quantum correction potential Δψ _q , the current is an auxiliary equation of the Poisson equation, the current continuity formula of electron concentration and hole concentration, and the current continuity formula Having a means for solving the density equation and obtaining self-consistent potential, electron concentration and hole concentration,
N in the relational expression of the quantum correction potential Δψ _q is a function satisfying the Schrödinger equation , the unit elementary charge q, the effective mass m ^* , the dielectric constant ε, the Planck constant h, and the depletion layer formed in the semiconductor substrate. B = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32) N], which is a function of the surface density N _depl of the impurity charge and the surface density N _inv of the electrons inverted in the semiconductor substrate. _inv ]] ^1/3 , the function satisfying the Schrödinger equation is expressed as Az ² exp (−bz) where A is the proportionality coefficient and z is the distance from the surface of the semiconductor substrate to the depth direction. It represented, the semiconductor device evaluation apparatus characterized that you the quantum correction potential is expressed as _{Δψ q (z) = βb (} b / 4-1 / z). A semiconductor element evaluation method for evaluating an insulated gate field effect transistor,
Solve Poisson's equation, which is the relational expression of potential, electron concentration and hole concentration,
By solving the Poisson equation, the potential is ψ, the unit charge is q, h ² / (24π ² qm ^* ) (where h is Planck's constant ) using the temporarily obtained potential, electron concentration and hole concentration. , M ^* is an effective mass, q is a unit elementary charge) , and a quantum correction potential coefficient represented by β and an electron concentration n is represented by a relational expression of Δψ _q = β∇ · ∇√n / √n. Quantum correction potential Δψ _q
Using the quantum correction potential Δψ _q temporarily obtained by calculating the quantum correction potential Δψ _q , the current is an auxiliary equation of the Poisson equation, the current continuity formula of electron concentration and hole concentration, and the current continuity formula A method for solving a density equation and obtaining self-consistent potential, electron concentration and hole concentration,
N in relations of the quantum correction potential [Delta] [phi] _q is Ri functions der satisfying Schrodinger equation, the unit elementary charge q, effective mass m ^*, the dielectric constant epsilon, the depletion layer formed Planck constant h, the semiconductor substrate B = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32)], which is a function of the surface density N _depl of the impurity charges and the surface density N _inv of the electrons inverted in the semiconductor substrate. N _inv ]] ^1/3 , the function satisfying the Schrödinger equation is represented by Az ² exp (−bz) where A is the proportionality coefficient and z is the distance from the surface of the semiconductor substrate in the depth direction. And the quantum correction potential is expressed as Δψ _q (z) = βb (b / 4/1 / z) . A computer-readable recording medium recording a semiconductor element evaluation program for evaluating an insulated gate field effect transistor,
A procedure for solving Poisson's equation which is a relational expression of potential, electron concentration and hole concentration;
Using the potential, electron concentration and hole concentration temporarily obtained by the procedure for solving the Poisson equation, the potential is ψ, the unit elementary charge is q, h ² / (24π ² qm ^* ) (where h is Planck's constant, Quantum correction represented by the relational expression Δψ _q = β∇ · ∇√n / √n, where β is the quantum correction potential coefficient represented by m ^* is effective mass, q is unit elementary charge) , and n is the electron concentration. A procedure for calculating the potential Δψ _q ;
Using the quantum correction potential Δψ _q temporarily obtained by the procedure of calculating the quantum correction potential Δψ _q , the current is an auxiliary equation of the Poisson equation, the current continuity equation of electron concentration and hole concentration, and the current continuity equation Having a procedure for solving the density equation and obtaining self-consistent potential, electron concentration and hole concentration,
N in relations of the quantum correction potential [Delta] [phi] _q is Ri functions der satisfying Schrodinger equation,
A function of unit elementary charge q, effective mass m ^* , dielectric constant ε, Planck's constant h, surface density N _depl of impurity charges in the depletion layer formed in the semiconductor substrate, and surface density N _inv of electrons inverted in the semiconductor substrate When b is expressed as b = [48π ² q ² m ^* / (εh ² ) [N _depl + (11/32) N _inv ]] ^1/3 , a function satisfying the Schrodinger equation has a proportional coefficient: A, where z is the distance in the depth direction from the surface of the semiconductor substrate, it is expressed as Az ² exp (−bz), and the quantum correction potential is Δψ _q (z) = βb (b / 4−1 / z) A computer-readable recording medium on which a semiconductor element evaluation program is recorded.

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