Methods of modeling a transistor and apparatus used therein

Methods of modeling a transistor are provided. The method includes the steps of (a) extracting reference mobility values of a channel layer of a transistor including a gate electrode, a source region and a drain region using a reference gate voltage, a reference drain current and a reference drain voltage, (b) fitting a mobility function including model parameters on the reference mobility values to extract the model parameters, and (c) putting the extracted model parameters into a drain current modeling function to calculate a drain current flowing through the channel layer between the drain region and the source region under a bias condition defined by an arbitrary gate voltage applied to the gate electrode and an arbitrary drain voltage applied to the drain region. Related apparatuses are also provided.

CROSS-REFERENCE TO RELATED APPLICATIONS

This U.S. non-provisional patent application claims priority under 35 U.S.C. §119 to Korean Patent Application No. 10-2011-0048064, filed on May 20, 2011, in the Korean Intellectual Property Office, the entire contents of which are hereby incorporated by reference herein.

BACKGROUND

1. Technical Field

The present disclosure herein relates to methods of modeling a transistor and apparatus used therein.

2. Description of Related Art

Semiconductor devices are widely used in various industrial areas such as electronic systems, automobiles and/or vessels because of light weight, small size, multi-function and/or low fabrication cost thereof. Field effect transistors (hereinafter, referred to as transistors) are important elements that constitute the semiconductor devices. In general, each of the transistors may include a gate electrode, a source region and a drain region. The source region and the drain region may be disposed in a semiconductor substrate and may be spaced apart from each other, and the gate electrode may be disposed on a channel region between the source region and the drain region. The source and drain regions may be formed by injecting dopant ions into the semiconductor substrate, and the gate electrode may be insulated from the channel region by a gate insulation layer. These transistors are widely used as switching elements constituting integrated circuits of the semiconductor device or single elements constituting logic circuits of the semiconductor device.

High reliable transistors have been increasingly demanded with the development of electronic industry. Accordingly, novel methods of modeling the transistors have been more required to improve the reliability of the transistors.

SUMMARY

Exemplary embodiments are directed to methods of modeling a high reliable transistor and apparatus used therein

According to some embodiments, a method of modeling a transistor includes the steps of (a) extracting reference mobility values of a channel layer of a transistor including a gate electrode, a source region and a drain region using a reference gate voltage, a reference drain current and a reference drain voltage, (b) fitting a mobility function including model parameters on the reference mobility values to extract the model parameters, and (c) putting the extracted model parameters into a drain current modeling function to calculate a drain current flowing through the channel layer between the drain region and the source region under a bias condition defined by an arbitrary gate voltage applied to the gate electrode and an arbitrary drain voltage applied to the drain region.

In an embodiment, the drain current modeling function may be expressed by the following equation,

Id=CWL⁢∫0Vd⁢(1Kb⁡(Vg-Von-Vy)β+1Ka⁡(Vg-Von-Vy)α)-1×(⁢Vg-Von-Vy)⁢ⅆVy
wherein, “Id” denotes the drain current, “Vg” denotes the arbitrary gate voltage, “Vd” denotes the arbitrary drain voltage, “L” denotes a channel length of the transistor, “W” denotes a channel width of the transistor, and “C” denotes a gate capacitance between the channel layer and the gate electrode.

In an embodiment, the method may further include additionally and repeatedly performing the step of extracting the reference mobility values and the step of extracting the model parameters against different channel layers at least once to extract a plurality of model parameter data sets, and storing a memory unit with the plurality of model parameter data sets.

In an embodiment, the method may further include selecting one set of the plurality of model parameter data sets. The drain current may be calculated using the selected of model parameter data set.

In an embodiment, the reference mobility values may be extracted using the following equation,

μr=LW⁢IrC⁡(Vg⁢⁢r-Von)⁢Vdr
wherein, “μr” denotes the reference mobility, “L” denotes a channel length of the transistor. “W” denotes a channel width of the transistor, “Ir” denotes the reference drain current, “C” denotes a gate capacitance between the channel layer and the gate electrode, “Vgr” denotes the reference gate voltage, “Von” denotes a turn on voltage of the transistor, and “Vdr” denotes the reference drain voltage.

In an embodiment, the reference drain voltage may be within the range of about 0.001 Volts to about 1.0 Volts.

In an embodiment, the method may further include applying the reference gate voltage and the reference drain voltage to the gate electrode and the drain region respectively to measure the reference drain current flowing through the channel layer between the drain region and the source region, prior to extraction of the reference mobility values.

In an embodiment, the transistor may include a thin film transistor.

In an embodiment, the channel layer may have an amorphous structure.

In an embodiment, the channel layer may include an organic semiconductor material or an oxide semiconductor material.

According to further embodiments, an apparatus used in modeling a transistor includes a reference mobility extraction unit, a fitting unit, and a drain current modeling unit. The reference mobility extraction unit extracts reference mobility values of a channel layer of the transistor including a gate electrode, a source region and a drain region using a reference gate voltage applied to the gate electrode and a reference drain current flowing through the channel layer in response to the reference gate voltage. The fitting unit extracts model parameters by fitting a mobility function of the model parameters on the reference mobility values. The drain current modeling unit calculates a drain current flowing through the channel layer by putting an arbitrary gate voltage applied to the gate electrode, an arbitrary drain voltage applied to the drain region and the model parameters into a drain current modeling function.

In an embodiment, the drain current modeling unit may additionally put a channel length of the transistor, a channel width of the transistor, a gate capacitance between the channel layer and the gate electrode, an arbitrary drain voltage applied to the drain region, and a turn on voltage of the transistor into the drain current modeling function.

In an embodiment, the fitting unit may extract the model parameters by fitting a mobility function on the reference mobility values.

The mobility function may be expressed by the following equation,

μ=(1Kb⁡(Vgate-Von)β+1Ka⁡(Vgate-Von)α)-1
wherein, “Ka”, “Kb”, “α” and “β” denote the model parameters, “Vgate” denotes the reference gate voltage corresponding to a gate voltage applied to the gate electrode, “Von” denotes a turn on voltage of the transistor, and “μ” denotes a mobility of the channel layer that is calculated using the model parameters, the reference gate voltage and the turn on voltage.

The drain current modeling function may be expressed by the following equation,

Id=CWL⁢∫0Vd⁢(1Kb⁡(Vg⁢-Von-Vy)β+1Ka⁡(Vg-Von⁢-Vy)α)-1×(Vg-Von-Vy)⁢ⅆVy
wherein, “Id” denotes the drain current, “Vg” denotes the arbitrary gate voltage, “Vd” denotes the arbitrary drain voltage, “L” denotes a channel length of the transistor, “W” denotes a channel width of the transistor, “C” denotes a gate capacitance between the channel layer and the gate electrode, and “Von” denotes the turn on voltage of the transistor.

In an embodiment, the number of the transistor may be two or more, the channel layers of the plurality of transistors may be formed of different material layers from each other, and the reference mobility extraction unit and the fitting unit may extract a plurality of model parameter sets including the model parameters.

In an embodiment, the apparatus may further include a memory unit storing the plurality of model parameter sets therein.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The advantages and features of the inventive subject matter and methods of achieving them will be apparent from the following exemplary embodiments that will be described in more detail with reference to the accompanying drawings. It should be noted, however, that the inventive subject matter is not limited to the following exemplary embodiments, and may be implemented in various forms. Rather, these exemplary embodiments are provided so that this disclosure will be thorough and complete, and will convey the scope of the disclosure to those skilled in the art.

It will be understood that when an element such as a layer, region or substrate is referred to as being “on” another element, it can be directly on the other, element or intervening elements may be present. In contrast, the term “directly” means that there are no intervening elements. In the drawings, the sizes and relative sizes of layers and regions may be exaggerated for clarity. It will be also understood that although the terms first, second, third etc. may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another element. Thus, a first element in some embodiments could be termed a second element in other embodiments without departing from the teachings of the present invention. Exemplary embodiments of aspects of the present inventive subject matter explained and illustrated herein include their complementary counterparts.

As used herein the term “and/or” includes any and all combinations of one or more of the associated listed items. The same reference numerals or the same reference designators denote the same elements throughout the specification.

FIG. 1is a flowchart illustrating a method of modeling a transistor according to an embodiment.

Referring toFIG. 1, a transistor for modeling may be provided. The transistor may be a thin film transistor formed on a substrate. The substrate may be a glass substrate or a plastic substrate. The substrate may be a flexible substrate. The transistor may include a gate electrode, a gate insulation layer, a source region, a drain region and a channel layer.

The channel layer may overlap with the gate electrode in a plan view. The gate insulation layer may be disposed between the channel layer and the gate electrode. According to some embodiments, the channel layer may be disposed above the gate electrode. Alternatively, the channel layer may be disposed below the gate electrode. When a voltage applied to the gate electrode is equal to or higher than a turn on voltage of the transistor, an inversion channel region may be formed in the channel layer.

The channel layer may include a semiconductor material. For example, the channel layer may include an organic semiconductor material or an oxide semiconductor material (e.g., a zinc oxide (ZnO) material or an indium-gallium-zinc oxide (IGZO) material). The channel layer may have an amorphous structure.

In a step S10, a reference gate voltage and a reference drain voltage are applied to the gate electrode of the transistor and the drain region of the transistor respectively, thereby measuring a reference drain current. While the reference drain current is measured, the source region may be grounded. The reference drain current may flows from the drain region toward the source region through the channel layer. The reference drain voltage may be within the range of about 0.001 Volts to about 1.0 Volts. If the reference drain voltage is fixed to have a constant value and the reference gate voltage varies, the reference drain current may also vary.

In a step S20, reference mobility values of the channel layer may be extracted using the reference gate voltages and the measured reference drain current values. Each of the reference mobility values may correspond to an average mobility of the channel layer. The reference mobility values may be extracted using the following equation 1. The equation 1:

In the equation 1, “μr” denotes the reference mobility of the transistor, and “L” denotes a channel length of the transistor. Further, “W” denotes a channel width of the transistor, and “Ir” denotes the reference drain current. Moreover, “C” denotes a gate capacitance between the channel layer and the gate electrode, and “Vgr” denotes the reference gate voltage. In addition, “Von” denotes a turn on voltage, and “Vdr” denotes the reference drain voltage.

In a step S30, a mobility function of the reference mobility and model parameters may be fitted on the reference mobility values extracted using the equation 1. As a result, the model parameters may be extracted.

When the transistor is a thin film transistor and the channel layer of the transistor has an amorphous structure, the mobility function may be divided into a tail state model function and a deep state model function. The tail state model function may be applied to the transistor when a Fermi level of the channel layer overlaps with a tail state adjacent to a conduction band of the channel layer, and the deep state model function may be applied to the transistor when a Fermi level of the channel layer overlap with a deep state around a mid gap between a conduction band and a valence band of the channel layer.

When the Fermi level of the channel layer overlaps with the deep state of the channel layer, the mobility of the channel layer may be proportional to a value expressed by (Vgate−Vfb)β. Here, “Vgate” denotes a gate voltage applied to the gate electrode of the transistor, and “Vfb” denotes a flat band voltage of the transistor. Further, “β” denotes a parameter that relates to a temperature characteristic at the deep state.

When the Fermi level of the channel layer overlaps with the tail state of the channel layer, the mobility of the channel layer may be proportional to a value expressed by (Vgate−Vt)α. Here, “Vt” denotes a threshold voltage of the transistor, and “α” denotes a parameter that relates to a temperature characteristic at the tail state.

The flat band voltage Vfb may be substantially equal to the turn on voltage of the transistor. In a thin film transistor employing an oxide semiconductor material as the channel layer, the threshold voltage Vt of the thin film transistor may be approximately equal to the turn on voltage when the gate voltage applied to the gate electrode is greater than a certain level.

If the gate voltage has a relatively low level, a relatively weak electric field may be applied to the channel layer. Accordingly, the mobility of the channel layer may be proportional to a value of (Vgate−Von)β. Alternatively, if the gate voltage has a relatively high level, a relatively strong electric field may be applied to the channel layer. Thus, the mobility of the channel layer may be proportional to a value of (Vgate−Von)α.

The mobility function used in the step S30may be expressed by a harmonic mean of the values of (Vgate−Von)βand (Vgate−Von)α, as described in the following equation 2.

The mobility function has the model parameters “Ka”, “Kb”, “α” and “β”. In the event that the equation 2 is fitted on the reference mobility values, the gate voltage Vgate in the equation 2 may correspond to the reference gate voltage Vgr. In the equation 2, “μ” may correspond to a mobility of the channel layer that is calculated using the model parameters, the reference gate voltage and the turn on voltage of the transistor. The model parameters “Ka”, “Kb”, “α” and “β” can be obtained by fitting the equation 2 on the reference mobility values extracted in the step S20.

In a step S40, the model parameter values “Ka”, “Kb”, “α” and “β” obtained in the step S30may be put into a drain current modeling function, thereby calculating a drain current that flows through the channel layer under a bias condition defined by an arbitrary gate voltage and an arbitrary drain voltage.

The drain current modeling function may be expressed by the following equation 3.

Wherein, “Id” denotes the drain current flows through the channel layer, “Vg” denotes the arbitrary gate voltage applied to the gate electrode, and “Vd” denotes the arbitrary drain voltage applied to the drain region.

Now, a modeling apparatus of a transistor according to an embodiment will be described.FIG. 2is a block diagram illustrating an apparatus used in modeling a transistor according to an embodiment.

Referring toFIG. 2, a modeling apparatus according to an embodiment may include a reference mobility extraction unit110, a fitting unit120and a drain current modeling unit130.

The reference mobility extraction unit110may receive reference gate voltage data (Data_Vgr), reference drain current data (Data_Ir), and transistor data (Data_Transistor) to extract reference mobility data (Data_μr). The extracted reference mobility data (Data_μr) may be output from the reference mobility extraction unit110.

The reference gate voltage data (Data_Vgr) may include data that relate to reference gate voltage levels applied to the gate electrode of the transistor. The reference drain current data (Data_Ir) may include data that relate to reference drain current values flowing through the channel layer between the drain region and the source region of the transistor. The reference drain current values of the reference drain current data (Data_Ir) may be measured by applying the reference gate voltage levels of the reference gate voltage data (Data_Vgr) to the gate electrode, as described with reference to the step S10.

The transistor data (Data_Transistor) may include a channel length L, a channel width W, a gate capacitance C, reference drain voltages, and a turn on voltage of the transistor.

The reference mobility data (Data_μr) may include data that relate to mobility values of the channel layer according to voltages applied to the gate electrode.

The reference mobility extraction unit110may extract the reference mobility values of the reference mobility data (Data_μr) using the reference gate voltage data (Data_Vgr), the reference drain current data (Data_Ir), the transistor data (Data_Transistor), and the equation 1 and may transmit the extracted reference mobility data (Data_μr) to the fitting unit120.

The fitting unit120may be configured to receive the reference mobility data (Data_μr) and to output a model parameter data set (Data_VARset). The fitting unit120may extract the model parameters “Ka”, “Kb”, “α” and “β” by fitting the equation 2 described in the step30on the reference mobility values of the reference mobility data (Data_μr). The model parameter data set (Data_VARset) including the model parameters “Ka”, “Kb”, “α” and “β” may be transmitted to the drain current modeling unit130.

The drain current modeling unit130may be configured to receive model parameter data set (Data_VARset), drain voltage data (Data_Vd), gate voltage data (Data_Vg) and the transistor data (Data_Transistor) and to output drain current data (Data_Id) of the transistor. The drain current modeling unit130may put the model parameters of the model parameter data set (Data_VARset) and the data of the transistor data (Data_Transistor) into the equation 3 described with reference to the step S40, thereby calculating drain current values that flows through the channel layer under bias conditions defined by arbitrary gate voltages of the gate voltage data (Data_Vg) and arbitrary drain voltages of the drain voltage data (Data_Vd).

A modeling apparatus of a transistor according to a modified embodiment will be described hereinafter.FIG. 3is a block diagram illustrating an apparatus used in modeling a transistor according to a modified embodiment. The modeling apparatus according to the present modified embodiment may be similar to the modeling apparatus according to the previous embodiment illustrated inFIG. 2. Thus, the following description will focus on differences between the present modified embodiment and the previously described embodiment ofFIG. 2.

Referring toFIG. 3, a modeling apparatus according to the present modified embodiment may include a reference mobility extraction unit110, a fitting unit120, a drain current modeling unit130and a memory unit140. The reference mobility extraction unit110, the fitting unit120and the drain current modeling unit130may have the same configurations as those described with reference toFIG. 2.

A modeled drain current of a first transistor may be obtained using the reference mobility extraction unit110, the fitting unit120and the drain current modeling unit130. That is, the modeled drain current of the first transistor may be obtained using the same manners as described with reference toFIGS. 1 and 2. A channel layer (hereinafter, referred to as a first channel layer) of the first transistor may be formed of a first material. A first model parameter data set (Data_VARset1) of the first transistor may be transmitted to the memory unit140and may be stored therein.

Subsequently, a modeled drain current of a second transistor may be obtained using the reference mobility extraction unit110, the fitting unit120and the drain current modeling unit130. That is, the modeled drain current of the second transistor may be also obtained using the same manners as described with reference toFIGS. 1 and 2. A channel layer (hereinafter, referred to as a second channel layer) of the second transistor may be formed of a second material different from the first material. A second model parameter data set (Data_VARset2) of the second transistor may be transmitted to the memory unit140and may be stored therein.

The above processes may be repeatedly performed to obtain a plurality of model parameter data sets, and the plurality of model parameter data sets may be stored in the memory unit140. That is, “n”-number of model parameter data sets (Data_VARset1˜Data_VARsetn,) of “n”-number of different transistors may be stored in the memory unit140(“n” is a natural number equal to or greater than two).

The memory unit140may receive a selection signal Sel, and the memory unit140may transmit any one the model parameter data set (Data_VARsetsel) selected from the group consisting of the plurality of model parameter data sets (Data_VARset1˜Data_VARsetn) to the drain current modeling unit130in response to the selection signal Sel. The drain current modeling unit130may produce a drain current data (Data_Id) of the corresponding transistor using the selected model parameter data set (Data_VARsetsel).

Various simulation results of the drain current obtained according to the embodiments will be described hereinafter.

FIG. 4is a graph illustrating a mobility characteristic of a transistor having an indium-gallium-zinc oxide (IGZO) channel layer obtained by fitting a mobility function on reference mobility values using a modeling method according to an embodiment.FIG. 5is a graph illustrating modeled transfer curves obtained using a modeling method according to an embodiment together with actually measured transfer curves, in conjunction with a transistor having an IGZO channel layer.FIG. 6is a graph illustrating modeled Id-Vd curves obtained using a modeling method according to an embodiment together with actually measured Id-Vd curves, in conjunction with a transistor having an IGZO channel layer.

Referring toFIG. 4, the abscissa denotes a reference gate voltage Vrg, and the ordinate denotes a reference mobility μrof a channel layer of a transistor. InFIG. 4, data indicated by reference numeral221correspond to reference mobility values of an IGZO channel layer relative to the reference gate voltage Vrg, which are extracted using the equation 1 described with reference to the steps S10and S20ofFIG. 1. Further, data indicated by reference designator221acorrespond to a mobility curve fitted on the reference mobility values relative to the reference gate voltage Vrg, which is obtained using the equation 2 described with reference to the steps S30ofFIG. 1. Model parameters extracted from the mobility curve221aofFIG. 4were listed in the following table 1.

Referring toFIG. 5, the abscissa denotes an arbitrary gate voltage Vg applied to the gate electrode of the transistor, and the ordinate denotes a drain current Id flowing through the channel layer between the drain region and the source region of the transistor. Transfer curves indicated by reference numerals222,223and224represent drain current data that are actually measured according to the gate voltage Vg. The transfer curves222,223and224were measured at drain voltages of about 0.3 Volts, about 3 Volts and about 30 Volts, respectively. Further, transfer curves indicated by reference designators222a,223aand224acorrespond to modeled transfer curves of the transistor obtained by putting the model parameters “Ka”, “Kb”, “α” and “β” extracted from the mobility curve221aofFIG. 4into the equation 3 described with reference to the step S40ofFIG. 1. The modeled transfer curves222a,223aand224awere obtained at the drain voltages of about 0.3 Volts, about 3 Volts and about 30 Volts, respectively.

Referring toFIG. 6, the abscissa denotes an arbitrary drain voltage Vd applied to the drain region of the transistor, and the ordinate denotes a drain current Id flowing through the channel layer between the drain region and the source region of the transistor. Id vs. Vd (Id-Vd) curves indicated by reference numerals225,226and227represent drain current data that are actually measured according to the drain voltage Vd. The Id-Vd curves225,226and227were measured at gate voltages of about 10 Volts, about 15 Volts and about 20 Volts, respectively. Further, Id-Vd curves indicated by reference designators225a,226aand227acorrespond to modeled Id-Vd curves of the transistor obtained by putting the model parameters “Ka”, “Kb”, “α” and “β” extracted from the mobility curve221aofFIG. 4into the equation 3 described with reference to the step S40ofFIG. 1. The modeled Id-Vd curves225a,226aand227awere obtained at the gate voltages of about 10 Volts, about 15 Volts and about 20 Volts, respectively.

As can be seen fromFIGS. 5 and 6, the transfer curves and the Id-Vd curves of the modeled transistor having the IGZO channel layer according to the embodiment were almost consistent with the measured transfer curves and the measured Id-Vd curves, respectively.

FIG. 7is a graph illustrating a mobility characteristic of a transistor having an amorphous silicon channel layer obtained by fitting a mobility function on reference mobility values using a modeling method according to an embodiment.FIG. 8is a graph illustrating modeled transfer curves obtained using a modeling method according to an embodiment together with actually measured transfer curves, in conjunction with a transistor having an amorphous silicon channel layer.FIG. 6is a graph illustrating modeled Id-Vd curves obtained using a modeling method according to an embodiment together with actually measured Id-Vd curves, in conjunction with a transistor having an amorphous silicon channel layer.

Referring toFIG. 7, the abscissa denotes a reference gate voltage Vrg, and the ordinate denotes a reference mobility μrof a channel layer of a transistor. InFIG. 7, data indicated by reference numeral231correspond to reference mobility values of an amorphous silicon channel layer relative to the reference gate voltage Vrg, which are extracted using the equation 1 described with reference to the steps S10and S20ofFIG. 1. Further, data indicated by reference designator231acorrespond to a mobility curve fitted on the reference mobility values relative to the reference gate voltage Vrg, which is obtained using the equation 2 described with reference to the steps S30ofFIG. 1. Model parameters extracted from the mobility curve231aofFIG. 7were listed in the following table 2.

Referring toFIG. 8, the abscissa denotes an arbitrary gate voltage Vg applied to the gate electrode of the transistor, and the ordinate denotes a drain current Id flowing through the channel layer between the drain region and the source region of the transistor. Transfer curves indicated by reference numerals232,233,234,235and236represent drain current data that are actually measured according to the gate voltage Vg. The transfer curves232,233,234,235and236were measured at drain voltages of about 0.1 Volts, about 0.5 Volts, about 2 Volts, about 5 Volts and about 10 Volts, respectively. Further, transfer curves indicated by reference designators232a,233a,234a,235aand236acorrespond to modeled transfer curves of the transistor obtained by putting the model parameters “Ka”, “Kb”, “α” and “β” extracted from the mobility curve231aofFIG. 7into the equation 3 described with reference to the step S40ofFIG. 1. The modeled transfer curves232a,233a,234a,235aand236awere obtained at the drain voltages of about 0.1 Volts, about 0.5 Volts, about 2 Volts, about 5 Volts and about 10 Volts, respectively.

Referring toFIG. 9, the abscissa denotes an arbitrary drain voltage Vd applied to the drain region of the transistor, and the ordinate denotes a drain current Id flowing through the channel layer between the drain region and the source region of the transistor. Id vs. Vd (Id-Vd) curves indicated by reference numerals237,238and239represent drain current data that are actually measured according to the drain voltage Vd. The Id-Vd curves237,238and239were measured at gate voltages of about 4 Volts, about 8 Volts and about 16 Volts, respectively. Further, Id-Vd curves indicated by reference designators237a,238aand239acorrespond to modeled Id-Vd curves of the transistor obtained by putting the model parameters “Ka”, “Kb”, “α” and “β” extracted from the mobility curve231aofFIG. 7into the equation 3 described with reference to the step S40ofFIG. 1. The modeled Id-Vd curves237a,238aand239awere obtained at the gate voltages of about 4 Volts, about 8 Volts and about 16 Volts, respectively.

As can be seen fromFIGS. 8 and 9, the transfer curves and the Id-Vd curves of the modeled transistor having the amorphous silicon channel layer according to the embodiment were almost consistent with the measured transfer curves and the measured Id-Vd curves, respectively.

According to the embodiments set forth above, a mobility function may be fitted on reference mobility values of a channel layer extracted using reference gate voltage values and reference drain current values, thereby extracting model parameters. The model parameters can be put into a drain current modeling function to calculate drain currents of the modeled transistor. Accordingly, high reliable and simple modeling methods of transistors can be provided.