JP6069918B2 - Simulation method and device simulator - Google Patents

Description

  The present invention relates to a simulation method and a device simulator for calculating a current-voltage characteristic of a device.

  In nanodevices, the local structure and interface greatly affect the properties of the entire device. Based on quantum mechanics, first-principles calculations that allow device properties to be calculated based on atomic type and position information only. is important.

  In recent years, the calculation of the current-voltage characteristics between the source and the drain by the nonequilibrium Green function method based on the first principle calculation has been actively performed. However, the amount of calculations for first-principles calculations is enormous, and calculations that actually incorporate the gate structure into the atomic structure model and apply the gate voltage are very expensive. For this reason, device characteristics when a gate voltage is applied are hardly investigated by a method based on first-principles calculations.

  For this reason, for example, in the calculation of the current-voltage characteristics between the source and the drain, an attempt has been made to calculate the gate electric field dependence by adding the model potential created by the gate electrode as an external field.

Taisuke Ozaki et al., "Efficient implementation of the nonequilibrium Green function method for electronic transport calculations", PHYSICAL REVIEW B 81, 035116 (2010), The American Physical Society, p. 035116-1 to 035116-19

  However, the induced charges generated by the potential applied as the external field described above do not correspond to reality, and there is a problem that the current-voltage characteristics between the source and the drain cannot be calculated with high accuracy.

  Therefore, an object of the present invention is to easily and practically incorporate a gate electric field corresponding to reality in the source-drain current-voltage characteristic calculation by the nonequilibrium Green function method based on the first-principles calculation. It is to calculate the voltage characteristics with high accuracy.

The disclosed technique is a computer-implemented simulation method that reads a structural model of a device stored in a storage unit, and calculates quantum current of a current flowing through a channel between a source and a drain by a nonequilibrium Green function method. An analytical potential produced by a surface charge having an end at the gate width of the gate that is infinitely long in one of the two directions perpendicular to the conduction direction and perpendicular to the other and opposite to the channel given by the structural model, The current is calculated by adding in a periodic manner in two directions perpendicular to the conduction direction.

  Further, as means for solving the above problems, a device simulator, a program, and a recording medium on which the program is recorded can be used.

  In the disclosed technique, the induced electric charge when a gate electric field corresponding to reality is applied is modeled, whereby the gate electric field dependence of the current-voltage characteristic can be calculated without greatly changing the calculation method.

It is a figure which shows a carbon chain structure model. It is a figure which shows the model potential of the gate by related technology. It is a figure which shows the induced charge by the model potential in related technology. It is a figure for demonstrating the induced charge by the model potential in related technology when the center area | region is lengthened. It is a figure for demonstrating the induced charge by the model potential in related technology when the area | region of Leads is lengthened. It is a schematic diagram of an infinitely long surface charge in one direction. It is a figure which shows the model potential of the gate by this Embodiment. It is a figure which shows the induced charge by the model potential of this Embodiment. It is a figure which shows the induced charge by the model potential of this Embodiment. It is a figure which shows the hardware constitutions of a device simulator. It is a figure which shows the function structural example of a device simulator. It is a figure which shows the example of a structural model. It is a flowchart figure for demonstrating the current-voltage characteristic calculation process in this Embodiment. It is a figure for demonstrating the calculation process of the electric current I in step S16 of FIG. It is a schematic diagram of the current-voltage characteristic of a device. It is a figure which shows the case where gate length is 16 atoms and chain is 32 atoms. It is a figure which shows the case of a double gate structure. It is a figure which shows the case where it is a double gate structure and there are two carbon chains. FIG. 18 is a diagram showing induced charges when the surface charge density is changed in the case of the double gate structure of FIG. 17.

  Hereinafter, embodiments of the present invention will be described with reference to the drawings. The inventor verified the charge induced by the model potential of the gate based on the description of Non-Patent Document 1 as a related technique.

  First, a carbon chain structure as shown in FIG. 1 will be described as an example. FIG. 1 is a diagram showing a carbon chain structure model. In the carbon structural model shown in FIG. 1, 2a represents a carbon atom, and the Reserver portion (regions 3a and 3b) composed of 4 carbon atoms repeats the same structure semi-infinitely. In this figure, this structure in which there are 12 carbon atoms 2a and 12 carbon atoms 2a are arranged in a line is called chain-12. In the Center + Leads region 4, the number of atoms changes depending on the size of the model, and the number of designation changes accordingly.

  According to Non-Patent Document 1, the potential applied in the calculation of the current-voltage characteristics of this structure is given by the following equation:

ここで、Ｘ_ｃは構造の中央の０．５であり、ｄは図１の場合、Center+Leadsの領域４の長さに相当する１／３である。また、Ｖ_ｇ ^（０）はポテンシャルの大きさを決める係数である。これを図示すると、図２のようになる。

Here, _Xc is 0.5 at the center of the structure, and d is 1/3 corresponding to the length of the Center + Leads region 4 in the case of FIG. V _g ⁽⁰⁾ is a coefficient that determines the magnitude of the potential. This is illustrated in FIG.

FIG. 2 is a diagram showing the model potential of the gate according to the related art. FIG. 2 shows a case where chain-12 and V _g ⁽⁰⁾ = 10 eV. Since this potential depends only on the x direction, which is the conduction direction, and is constant in the y and z directions perpendicular to it, it is a non-equilibrium Green function based on first-principles calculations that treat the y and z directions with periodic boundary conditions. Can be easily incorporated into the law. This is because the two-way Poisson equation can be calculated using the fast Fourier transform.

  The inventor calculated the induced charge created by this potential, however, it was as shown in FIG. FIG. 3 is a diagram showing the induced charge due to the model potential in the related art. The horizontal axis represents each carbon atom, and the vertical axis represents the charge amount of each atom, representing the difference from when there is no gate potential.

  In FIG. 3, there is a large induced charge 5 a at the end of the Center + Leads region 4, and a large induced charge 5 b having an opposite sign is present just outside. Although the charge is induced, it can be seen that it is difficult to deal with reality.

  In addition, the inventor verified the induced charge generated by the potential when the Center region and the Leads region are lengthened in the Center + Leads region 4. FIG. 4 is a diagram for explaining the induced charges due to the model potential in the related art when the Center region is lengthened. Fig. 4 (A) shows the model potential when the center region is lengthened with chain-28, and Fig. 4 (B) shows the induced charge calculated based on the model potential shown in Fig. 4 (A). ing.

  When the center region is lengthened, as shown in FIG. 4B, in the chain-28 carbon chain structure, the induced charge is large at both ends (Leads portion) in the center + Leads region 4, It was shown that the induced charge was small, and a large amount of negative induced charge was generated just outside the region 4.

  FIG. 5 is a diagram for explaining the induced charges due to the model potential in the related art when the Leads region is lengthened. 5A shows the model potential when the Leads region is lengthened with chain-28, and FIG. 5B shows the induced charge calculated based on the model potential shown in FIG. 5A. ing.

  When the Leads region is lengthened, as shown in FIG. 5 (B), in the chain-28 carbon chain structure, there is a large induced charge at the boundary between Center and Leads in the Center + Leads region 4. As a result, an induced charge with a large reverse sign was generated immediately outside.

  As described above, it has been found that even if the center area is lengthened or the leads area is lengthened, there is no improvement.

  In this embodiment, the inventor can simulate the generation of more realistic induced charges by introducing a model potential for periodic gate application in two directions perpendicular to the conduction direction. The source / drain current-voltage characteristics can be calculated with high accuracy. Further, in this embodiment, when the atomic structure is not continuous in the direction perpendicular to the conduction direction and a vacuum layer exists, the potential may not be continuous as long as the potential is periodic. You just have to be there.

  FIG. 6 is a schematic diagram of an infinitely long surface charge in one direction. The surface charge illustrated in FIG. 6 is a model of the induced charge when a gate electric field corresponding to reality is applied, and a indicates half of the width of the surface charge (corresponding to the gate length). R represents a reference point of potential and may be used anywhere. The inventor has analytically shown that the potential generated by such a surface charge having an infinite unit charge density in one direction is given by the following equation.

ここで、ｐ（ｘ，ｚ）がポテンシャル、εは面電荷の周りの材質の誘電率である。

Here, p (x, z) is the potential, and ε is the dielectric constant of the material around the surface charge.

Here, the carbon chain structure will be described as an example. In the case of the structure shown in FIG. 1, a = 1/6, and R is now set at the points X = 0 and Z = 0. It is assumed that the carbon chain is at y = 0 and z = 0. The above equation is illustrated in FIG. Here, the charge density of the surface charge was 10 ¹³ cm ⁻² .

  FIG. 7 is a diagram showing the model potential of the gate according to the present embodiment. FIG. 7 shows the case of chain-12. Unlike Non-Patent Document 1 as a related technology, it also has dependency in the z direction. Now, the surface charge is illustrated as being at z = −1 and is not included in the region of FIG. This potential may be repeated periodically in the z direction, and since there is no dependence in the y direction, it can be handled periodically.

  When the atomic structure is not continuous in the direction perpendicular to the conduction direction and a vacuum layer is present, the potential may not be continuous as long as the potential is periodic, and the jump of the potential may be just in the vacuum layer.

  Since the atomic column is now at the position of z = 0, this makes it possible to calculate the source / drain current voltage characteristics taking into account the influence from the surface charge at a distance of z = -1. Specifically, in this model, it is possible to simulate a realistic nanodevice using a gate electrode having a 1 nm vacuum as an insulating layer.

  The charge induced by this potential is as shown in FIG. 8 and is similar to FIG. FIG. 8 is a diagram showing the induced charges due to the model potential of the present embodiment. FIG. 8 shows the induced charges of each atom in the case of chain-12. However, in the case of the model potential of the present invention, it has been found that the realistic charge distribution can be reproduced as shown in FIG.

  FIG. 9 is a diagram showing the induced charges due to the model potential of the present embodiment. FIG. 9 shows the induced charge of each atom in the case of chain-28 with a long lead portion. As shown in FIG. 9, in the case of chain-28, a curve is drawn with the highest difference in the center portion with no gate potential, and the difference becomes smaller toward both ends (x = 0 and x = 1). .

  In the present embodiment, it is possible to obtain a realistic calculation result of induced charges only by using a carbon chain structure with a long lead portion, and therefore, it is possible to calculate a current value with higher accuracy.

  A device simulator according to the present embodiment will be described. FIG. 10 is a diagram illustrating a hardware configuration of the device simulator. In FIG. 10, a device simulator 100 is a terminal controlled by a computer, and includes a CPU (Central Processing Unit) 11, a main storage device 12, an auxiliary storage device 13, an input device 14, a display device 15, The output device 16, a communication I / F (interface) 17, and a drive 18 are connected to the bus B.

  The CPU 11 controls the device simulator 100 according to a program stored in the main storage device 12. The main storage device 12 uses a RAM (Random Access Memory) or the like, and stores a program executed by the CPU 11, data necessary for processing by the CPU 11, data obtained by processing by the CPU 11, and the like. A part of the main storage device 12 is allocated as a work area used for processing by the CPU 11.

  The auxiliary storage device 13 uses a hard disk drive and stores data such as programs for executing various processes. A part of the program stored in the auxiliary storage device 13 is loaded into the main storage device 12 and executed by the CPU 11, whereby various processes are realized. The storage unit 130 includes the main storage device 12 and / or the auxiliary storage device 13.

  The input device 14 includes a mouse, a keyboard, and the like, and is used for a user to input various information necessary for processing by the device simulator 100. The display device 15 displays various information required under the control of the CPU 11. The output device 16 has a printer or the like and is used for outputting various types of information in accordance with instructions from the user. The communication I / F 17 is a device that is connected to, for example, the Internet, a LAN (Local Area Network), etc., and controls communication with an external device.

  A program that realizes processing performed by the device simulator 100 is provided to the device simulator 100 by a storage medium 19 such as a CD-ROM (Compact Disc Read-Only Memory). That is, when the storage medium 19 storing the program is set in the drive 18, the drive 18 reads the program from the storage medium 19, and the read program is installed in the auxiliary storage device 13 via the bus B. . When the program is activated, the CPU 11 starts its processing according to the program installed in the auxiliary storage device 13. The medium for storing the program is not limited to a CD-ROM, and any medium that can be read by a computer may be used. As a computer-readable storage medium, in addition to a CD-ROM, a portable recording medium such as a DVD disk or a USB memory, or a semiconductor memory such as a flash memory may be used.

  FIG. 11 is a diagram illustrating a functional configuration example of the device simulator. In FIG. 11, the device simulator 100 includes an atomic coordinate input unit 21, a gate data input unit 22, and a current-voltage characteristic calculation unit 23 as processing units performed by the CPU 11. Each processing unit 21 to 23 is realized by the CPU 11 executing a corresponding program.

  The storage unit 130 stores atomic coordinate data 31, gate data 32, and calculation results 33. The structural model 7m (FIG. 12) is expressed by the atomic coordinate data 31 and the gate data 32.

  The atomic coordinate input unit 21 is a processing unit that inputs atomic coordinate data 31 from the storage unit 130. The gate data input unit 22 is a processing unit that inputs the gate data 32 from the storage unit 130. The current-voltage characteristic calculation unit 23 is a processing unit that calculates a current-voltage characteristic based on the structural model 7 m obtained from the atomic coordinate data 31 and the gate data 32. The current-voltage characteristic calculator 23 generates an analytical potential (Equation 2) generated by an infinitely long surface charge (FIG. 6) in one direction in the quantum conduction calculation by the nonequilibrium Green function method based on the first principle calculation. In addition to two directions (y direction and z direction) perpendicular to the (x direction) in a periodic manner (periodically infinite in the y direction), the current-voltage characteristics are obtained.

FIG. 12 shows an example of a structural model. 12, the structural model 7 has a gate 7 g, and the source 7s, and the channel 7c, and the drain 7d, and the gate voltage _{V G,} and a source-drain voltage _{V SD.}

  In this embodiment, the gate 7g is expressed by an infinitely long surface charge in one direction shown in FIG. 6, and the surface charge is applied to the channel 7c by using the above equation 2 that analytically shows the potential generated by the surface charge. The current I between the source 7s and the drain 7d in consideration of the influence is calculated.

  FIG. 13 is a flowchart for explaining the current-voltage characteristic calculation processing in the present embodiment. The current-voltage characteristic calculation process shown in FIG. 11 is realized by the CPU 11 executing a corresponding program.

  In FIG. 11, the atomic coordinate input unit 21 activated by the control of the CPU 11 reads and inputs the atomic coordinate data 31 from the storage unit 130 (step S11). The atomic coordinate input unit 21 may read the atomic coordinate data 31 stored in the storage unit 130 in advance, or displays a predetermined user interface for acquiring the atomic coordinate data 31 on the display device 15 and acquires it from the user. You may make it do. The atomic coordinate data 31 acquired from the user is stored in the storage unit 130. The atomic coordinate data 31 includes atomic coordinates related to the source, drain, and channel.

  Next, the CPU 11 activates the gate data input unit 22 to read and input the gate data 32 from the storage unit 130 (step S12). The gate data input unit 22 may read the gate data 32 stored in the storage unit 130 in advance, or may display a predetermined user interface for acquiring the gate data 32 on the display device 15 and acquire it from the user. Anyway. The gate data 32 acquired from the user is stored in the storage unit 130. The gate data 32 includes values such as the number of gates, position, and gate width. The length of the leads is based on the gate width.

Then, the CPU 11 activates the current-voltage characteristic calculation unit 23. The current-voltage characteristic calculator 23 calculates the gate voltage V _GO (r) (step S13). In the calculation of the gate voltage V _GO (r) in step S13, the gate 7g is represented by an infinitely long surface charge in one direction shown in FIG. 6, and is based on the above equation 2 that analytically shows the potential generated by the surface charge. Is calculated.

The current-voltage characteristic calculator 23 loops for each gate voltage V _G (r) (step S14) and calculates the current I according to the source / drain voltage V _SD (steps S15 and S16). A calculation result 33 by the current-voltage characteristic calculation unit 23 is stored in the storage unit 130.

FIG. 14 is a diagram for explaining the calculation process of the current I in step S16 of FIG. In FIG. 14, the current-voltage characteristic calculator 23 initializes a variable i to 0 and calculates the charge density ρ ₀ (r) (step S31).

The current-voltage characteristic calculation unit 23 increments the variable i by 1 (step S32), and calculates V _i (r) based on the charge density ρ calculated immediately before (step S33). Then, the current-voltage characteristic calculation unit 23 calculates V _i (r) + V _G V _G0 (r) (step S34). The current-voltage characteristic calculator 23 calculates ρ _i (r) based on the voltage calculated in step S34 (step S35).

It is determined whether or not the difference between ρ _i (r) and ρ _i-1 (r) has converged below a predetermined value Δ (step S36). If not converged, the voltage characteristic calculator 23 returns to step S32 and repeats the same processing as described above until the value of the charge density converges.

On the other hand, when the voltage converges, the voltage characteristic calculation unit 23 calculates the transmittance T (E) (step S37), and calculates the current I by integrating the transmittance T (E) (E = 0 to V _SD ). (Step S38).

FIG. 15 illustrates the current-voltage characteristic of the device obtained by the current-voltage characteristic calculation process as described above. FIG. 15 is a schematic diagram of the current-voltage characteristics of the device. FIG. 15 shows current-voltage characteristics when gate voltages V _G (1), V _G (2), and V _G (3) are applied.

Hereinafter, examples of device simulation by the current-voltage characteristic calculation processing according to the present embodiment will be shown. In FIG. 9, the gate length (length represented by a in FIG. 6) is a length of 4 atoms, but by increasing the leads portion according to the gate width, various gate lengths can be obtained. Can be calculated.
[Example 1]
FIG. 16 shows a case where the gate length is 16 atoms and the chain is 32 atoms. FIG. 16 shows a case where the gate length is 16 atoms. FIG. 16A shows the model potential and FIG. 16B shows the induced charge when the gate length is 16 atoms and the chain is 32 atoms. The lead length may be longer than this, and the induced charge corresponds to the actual induced charge and shows the same result.
[Example 2]
FIG. 17 is a diagram showing a case of a double gate structure. In FIG. 17, it is assumed that there is a surface charge on the opposite side of the carbon chain, and the above-described potential of 2 is overlapped to simulate a double gate structure. FIG. 17A shows the model potential and FIG. 17B shows the induced charge in such a double gate structure. It can be seen that approximately twice as much charge is induced as compared with FIG. 16 due to the double gate effect.
[Example 3]
In the case of the double gate structure of FIG. 17, it is possible to widen the space between the surface charges and arrange two carbon chains. It can be seen that the same charge is induced in the two carbon chains, and this method can be applied even if the channel layer has a thickness, that is, has two or more carbon chains in the thickness direction.

FIG. 18 is a diagram showing a double gate structure and two carbon chains. FIG. 18 shows a double gate structure in which the gate length is 16 atoms and the chain has 2 32 atoms. FIG. 18A shows the model potential and FIG. 18B shows the induced charge in such a double gate structure. FIG. 18B shows induced charges of two chains.
[Example 4]
In the case of the double gate structure of Example 2, the surface charge density was 10 ¹³ cm ⁻² . By changing the charge density, the charge induced accordingly can be controlled. FIG. 19 is a diagram showing the induced charges when the surface charge density is changed in the case of the double gate structure of FIG. In FIG. 19, the numerical value indicates the amount of charge induced in the central atom of the chain.

  As described above, by using the current-voltage characteristic calculation processing according to the present embodiment, the current-voltage characteristics corresponding to the device structure such as the number of gates, the gate length, the length of the carbon chain, and the number of the gates can be calculated. .

  In the calculation of the current-voltage characteristics between the source and drain of a nanodevice, the gate is modeled by a surface charge with an infinite length in one direction and an edge in the other direction, and a realistic charge distribution is reproduced by the potential created by the surface charge. Therefore, the current can be calculated with high accuracy.

  Therefore, according to the present embodiment, in the calculation of the current-voltage characteristics between the source and the drain by the nonequilibrium green function method based on the first principle calculation, the calculation amount is greatly increased without largely changing the method. Therefore, it is possible to model the induced charge when a gate electric field corresponding to reality is applied simply and easily, and the gate electric field dependence of the current-voltage characteristics can be calculated.

  In the present embodiment, the target for calculating the current-voltage characteristics is the carbon chain, but the present invention is not limited to this.

  The present invention is not limited to the specifically disclosed embodiments, and various modifications and changes can be made without departing from the scope of the claims.

The following additional notes are further disclosed with respect to the embodiment including the above examples.
(Appendix 1)
A simulation method executed by a computer,
Read the device structural model stored in the storage unit,
In the quantum conduction calculation by the nonequilibrium Green's function method, the analytical potential generated by the surface charge that is infinite in one direction and has an edge with the gate width given by the above structure model is expressed in two directions perpendicular to the conduction direction. A simulation method characterized by calculating a current in addition to a periodic form.
(Appendix 2)
The simulation method according to appendix 1, wherein the potential is periodic but not continuous.
(Appendix 3)
The simulation method according to appendix 2, wherein a vacuum layer is provided in the atomic structure represented by the structural model, and the jump is present in the vacuum layer.
(Appendix 4)
4. The simulation method according to claim 1, wherein the potential is overlapped by two or more surface charges.
(Appendix 5)
The simulation method according to claim 4, wherein the structural model has an atomic structure of a thick conductive layer in a flat region of the potential.
(Appendix 6)
A storage unit that stores a structural model of the device;
When performing quantum conduction calculation by the nonequilibrium Green's function method, an analytical charge is generated that has an infinite length in one direction and a surface charge having an edge at the gate width given by the structural model stored in the storage unit on the other side. A device simulator comprising: a characteristic calculation unit that calculates a current by adding a potential in two directions perpendicular to a conduction direction in a periodic manner.
(Appendix 7)
Read the device structural model stored in the storage unit,
In the quantum conduction calculation by the nonequilibrium Green's function method, the analytical potential generated by the surface charge that is infinite in one direction and has an edge with the gate width given by the above structure model is expressed in two directions perpendicular to the conduction direction. A simulation method characterized by calculating a current in addition to a periodic form.

7c channel 7d drain 7g gate 7s source 7m structural model 11 CPU
12 Main storage device 13 Auxiliary storage device 14 Input device 15 Display device 16 Output device 17 Communication I / F
18 Drive 19 Storage medium 21 Atomic coordinate input unit 22 Gate data input unit 23 Current voltage characteristic calculation unit 31 Atomic coordinate data 32 Gate data 33 Calculation result 100 Device simulator 130 Storage unit

Claims (5)

A simulation method executed by a computer,
Read the device structural model stored in the storage unit,
In quantum conduction calculation of current flowing through the channel between the source and the drain by nonequilibrium Green function method is given by a vertical the structural model to the other The endless length in one direction of the two directions perpendicular to the conduction direction channel A simulation method characterized in that a current is calculated by adding an analytical potential generated by a surface charge having an edge with a gate width of a gate opposite to a gate in a periodic form in two directions perpendicular to the conduction direction.   The simulation method according to claim 1, wherein the potential is periodic but not continuous.   The simulation method according to claim 2, wherein a vacuum layer is provided in the atomic structure represented by the structural model, and the jump is present in the vacuum layer.   4. The simulation method according to claim 1, wherein the potential is overlapped by two or more surface charges. A storage unit that stores a structural model of the device;
When performing the quantum conduction calculation of the current flowing through the channel between the source and drain by the nonequilibrium Green function method , one of the two directions perpendicular to the conduction direction is infinitely long in one direction, and the other is stored in the storage unit. A characteristic for calculating a current by adding an analytical potential generated by a surface charge having an edge at the gate width of the gate facing the channel given by the structural model in two directions perpendicular to the conduction direction. A device simulator comprising a calculation unit.

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