JP3247367B2 - Method for evaluating characteristics of semiconductor device - Google Patents
Method for evaluating characteristics of semiconductor deviceInfo
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- JP3247367B2 JP3247367B2 JP23008790A JP23008790A JP3247367B2 JP 3247367 B2 JP3247367 B2 JP 3247367B2 JP 23008790 A JP23008790 A JP 23008790A JP 23008790 A JP23008790 A JP 23008790A JP 3247367 B2 JP3247367 B2 JP 3247367B2
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Description
【発明の詳細な説明】 [発明の目的] (産業上の利用分野) 本発明は、半導体素子の電気特性を解析評価する手法
に係わり、特に電子及び正孔のエネルギー輸送効果を含
めて解析評価する半導体素子の特性評価方法に関する。DETAILED DESCRIPTION OF THE INVENTION [Object of the Invention] (Industrial application field) The present invention relates to a method for analyzing and evaluating the electrical characteristics of a semiconductor device, and particularly relates to an analysis and evaluation including an energy transport effect of electrons and holes. The present invention relates to a method for evaluating characteristics of a semiconductor device to be manufactured.
(従来の技術) 半導体素子内部のエネルギー輸送効果を含めて素子の
電気特性を解析評価する方法としては、例えば、「エネ
ルギー輸送モデルによるホット・キャリア解析」(片山
弘造他,電子情報通信学会技術研究報告CAS86−93,1986
年)がある。この技術は、第7図に示すように半導体内
部に離散化用の格子点を設け、各格子点の上でポアソン
方程式,電子電流方程式,正孔電流方程式,電子エネル
ギー保存式及び正孔エネルギー保存式の合計5つの連立
偏微分方程式を、未知数である電位分布ψ,電子濃度分
布n,正孔濃度分布p,電子エネルギー分布wE,電子エネル
ギー分布WHについて解くものであり、通常は大型コンピ
ュータ上で数値的に解くプログラムの形で実現される。
なお、第7図中701はp型半導体基板、702はn+拡散層
(ソース)、703はn+拡散層(ドレイン)、704は酸化
膜、705はソース電極、706はゲート電極、707はドレイ
ン電極を示している。(Prior art) As a method of analyzing and evaluating the electrical characteristics of a device including the energy transport effect inside a semiconductor device, for example, “Hot carrier analysis by energy transport model” (Katayama Kozo et al., IEICE technical research) Report CAS86-93,1986
Year). In this technique, as shown in FIG. 7, a discretization lattice point is provided inside a semiconductor, and a Poisson equation, an electron current equation, a hole current equation, an electron energy conservation equation, and a hole energy conservation equation are provided on each lattice point. This solves a total of five simultaneous partial differential equations for the unknown potential distribution ψ, electron concentration distribution n, hole concentration distribution p, electron energy distribution w E , and electron energy distribution W H , which are usually unknown. It is implemented in the form of a program that solves numerically above.
In FIG. 7, 701 is a p-type semiconductor substrate, 702 is an n + diffusion layer (source), 703 is an n + diffusion layer (drain), 704 is an oxide film, 705 is a source electrode, 706 is a gate electrode, and 707 is 5 shows a drain electrode.
上記5つの連立偏微分方程式は、以下のように定義さ
れる。The above five simultaneous partial differential equations are defined as follows.
▽・(ε▽ψ)=−q(p−n+ND−NA …(1) 但し、ε:半導体の誘電率、q:単位素電荷、t:時刻、
▽:空間微分演算子、ND:ドナー型イオン化不純物濃
度、NA:アククセプタ型イオン化不純物濃度、μE:電子
移動度、μH:正孔移動度、 :電子速度ベクトル、 :正孔速度ベクトル、kB:ボルツマン定数、TE:電子温
度、TH:正孔温度、GR:電子及び正孔の単位時間・単位体
積当りの生成消滅率、τWE:電子エネルギー緩和時間、
τWH:正孔エネルギー緩和時間、である。▽ · (ε ▽ ψ) = - q (p-n + N D -N A ... (1) Where ε: dielectric constant of semiconductor, q: unit elementary charge, t: time,
▽: spatial differential operator, N D : donor-type ionized impurity concentration, N A : acceptor-type ionized impurity concentration, μ E : electron mobility, μ H : hole mobility, : Electron velocity vector, : Hole velocity vector, k B : Boltzmann constant, T E : electron temperature, T H : hole temperature, GR: extinction rate of electron and hole per unit time and unit volume, τ WE : electron energy relaxation time ,
τ WH : hole energy relaxation time.
なお、wE,wH,TE,THの間には、以下の関係がある。Note that the following relationship exists between w E , w H , T E , and T H.
但し、mE:電子の有効質量、mH:正孔の有効質量、であ
る。 Here, m E is the effective mass of electrons, and m H is the effective mass of holes.
前記の5つの方程式(1)〜(5)は、未知数である
ψ,n,p,wE,wHに関する非線形方程式であるため、物理的
な考察から定めた初期解を基にして反復的に解を求める
手法が用いられる。Since the above-mentioned five equations (1) to (5) are nonlinear equations related to unknowns 数, n, p, w E and w H , iterative repetition is performed based on an initial solution determined from physical considerations. A method for finding a solution is used.
エネルギー保存式(4)(5)の左辺第3項は、電子
及び正孔にエネルギーを供給する電力密度項である。従
来法では、この項を見積る際、第8図のように局所的な
電位勾配▽ψと速度 或いは を用いて、その内積を電力項としていた。このことを電
子について模式的に表すと第9図のようになる。即ち、
十分長い距離の間に電位勾配▽ψと速度 がほぼ一定とみなせる場所において、エネルギー緩和時
間τWEの間に電子が走行する距離は である。その間に得るエネルギーq△ψをτWEで割った
ものが、電子が単位時間当り電位勾配から得るエネルギ
ー、即ち電力である。その電力に電子濃度nを掛けると
(4)式の左辺第3項の電力密度項になる。正孔につい
ても同様である。The third term on the left side of the energy conservation equations (4) and (5) is a power density term for supplying energy to electrons and holes. In the conventional method, when estimating this term, as shown in FIG. Or And the inner product was used as the power term. This is schematically shown in FIG. 9 for electrons. That is,
Potential gradient ▽ ψ and velocity over a sufficiently long distance Where the electron travels during the energy relaxation time τ WE It is. The energy obtained during that time divided by τ WE is the energy that electrons obtain from the potential gradient per unit time, that is, the power. When the power is multiplied by the electron concentration n, the power density term of the third term on the left side of the equation (4) is obtained. The same applies to holes.
ところが、従来技術の電力の計算方法は、十分長い距
離の間に電位勾配▽ψと速度 がほぼ一定とみなせる場合にのみ適用可能である。非常
に微細な半導体素子内部において、電位ψ及び電位勾配
▽ψの変動が、第10図のように 以下の距離で発生する場合、従来技術では の間に得るエネルギーが第10図中の△ψ1となり、実際
に電子が得られるエネルギー△ψ2よりも大きくなって
しまう。従って、エネルギーの供給源である電力項を過
大評価するので、最終的に得られるエネルギー分布も過
大評価することになる。これは、半導体素子の電気的特
性を誤って予測することにつながる。以上のことは、物
理的精度が最も高い粒子シミュレータによる解析によっ
ても指摘されている(例えば、M.Tomizawa,「Nonstatio
nary Carrier Dynamics in Quarter−Micron Si−MOS−
FET'S」,IEEE TRANSACTION ON COMPUTER−AIDED DESIG
N,VOL.7,NO.2,1988年)。However, the conventional power calculation method uses the potential gradient ▽ ψ and the speed over a sufficiently long distance. Is applicable only if can be considered almost constant. The fluctuation of the potential 内部 and the potential gradient に お い て inside a very fine semiconductor element is as shown in FIG. If it occurs at the following distance, △ [psi 1 next to the energy in FIG. 10 to obtain between, would actually larger than the energy △ [psi 2 electrons are obtained. Therefore, since the power term as the energy supply source is overestimated, the finally obtained energy distribution is also overestimated. This leads to erroneous prediction of the electrical characteristics of the semiconductor device. The above is also pointed out by analysis using a particle simulator with the highest physical accuracy (for example, M. Tomizawa, “Nonstatio
nary Carrier Dynamics in Quarter−Micron Si−MOS−
FET'S '', IEEE TRANSACTION ON COMPUTER-AIDED DESIG
N, VOL. 7, NO. 2, 1988).
(発明が解決しようとする課題) このように従来、エネルギー輸送を含めた特性評価方
法では、非常に微細な素子等の解析において、エネルギ
ー緩和時間の間に電子や正孔が走行する距離の間で、エ
ネルギーを供給する電位勾配が急峻に変化する場合、得
られるエネルギー分布を過大評価し、誤った特性を予測
する問題があった。(Problems to be Solved by the Invention) As described above, conventionally, in the characteristic evaluation method including the energy transport, in the analysis of a very fine element and the like, the distance over which the electrons and holes travel during the energy relaxation time is considered. In the case where the potential gradient for supplying energy changes steeply, there is a problem in that the obtained energy distribution is overestimated and erroneous characteristics are predicted.
本発明は、上記事情を考慮してなされたもので、その
目的とするところは、微小距離で電位勾配が変動する場
合にも物理的に妥当なエネルギーが得られるように、エ
ネルギー保存式の電力項又は電力密度項を物理的な考察
から合理的に見積ることができ、特性予測の物理的精度
の高い半導体素子の特性評価方法を提供することにあ
る。The present invention has been made in consideration of the above circumstances, and an object thereof is to provide an energy-conserving electric power so that a physically appropriate energy can be obtained even when a potential gradient fluctuates at a minute distance. It is an object of the present invention to provide a method for evaluating the characteristics of a semiconductor device, which is capable of rationally estimating a term or a power density term from physical considerations and has high physical accuracy of characteristic prediction.
[発明の構成] (課題を解決するための手段) 本発明の骨子は、半導体素子内部のエネルギー輸送効
果を含めて素子の電気的特性を解析する評価方式におい
て、エネルギー保存式の電力項を見積る際、エネルギー
緩和時間の間に走行する始点と終点の間の電位差からエ
ネルギー緩和時間内に得られるエネルギーを求めること
を特徴とする。[Summary of the Invention] The gist of the present invention is to estimate a power term of an energy-conserving equation in an evaluation method for analyzing electric characteristics of an element including an energy transport effect inside a semiconductor element. In this case, energy obtained within the energy relaxation time is obtained from the potential difference between the start point and the end point traveling during the energy relaxation time.
即ち本発明は、半導体素子内の電位分布,電子濃度分
布及び正孔濃度分布の内の少なくとも一つの物理量を求
め、且つ電子と正孔のエネルギー分布の少なくとも一方
を求めるために、ポアソン方程式,電子電流連続式,正
孔電流連続式,電子エネルギー保存式及び正孔エネルギ
ー保存式を解くことによって、半導体素子の電気的特性
を評価する方法において、前記エネルギー保存式中の単
位時間当りのエネルギー供給項である電力項又は電力密
度項を定める際、電子又は正孔がエネルギーを保持する
平均時間であるエネルギー緩和時間の間に、電子又は正
孔が走行する始点と終点を検出し、該検出された始点と
終点の間の電位差をエネルギー緩和時間で割って終点に
おける電力項又は電力密度項とするようにした方法であ
る。That is, the present invention provides a Poisson's equation, an electron, and a method for determining at least one physical quantity among a potential distribution, an electron concentration distribution, and a hole concentration distribution in a semiconductor element, and determining at least one of the energy distribution of electrons and holes. A method for evaluating the electrical characteristics of a semiconductor device by solving a continuous current type, a continuous hole current type, an electron energy conservation type, and a hole energy conservation type, wherein an energy supply term per unit time in the energy conservation type When determining the power term or the power density term, during the energy relaxation time, which is the average time for electrons or holes to retain energy, the starting point and the end point where electrons or holes travel are detected, and the detected This is a method in which the potential difference between the start point and the end point is divided by the energy relaxation time to obtain a power term or a power density term at the end point.
(作用) 本発明によれば、エネルギー輸送効果を含めて半導体
素子の電気的特性の解析評価において、エネルギー保存
式の電力項を見積る際、エネルギー緩和時間の間に走行
する始点と終点の間の電位差からエネルギー緩和時間内
に得られるエネルギーを求めるようにしている。従っ
て、エネルギーを供給する電位勾配が、エネルギー緩和
時間に走行する距離以内で大きな変動をしても、物理的
に妥当なエネルギー分布を求めることができるので、従
来技術よりも特性予測の物理的精度が高い。(Action) According to the present invention, when estimating the power term of the energy conservation equation in the analysis and evaluation of the electrical characteristics of the semiconductor device including the energy transport effect, the time between the start point and the end point traveling during the energy relaxation time is estimated. The energy obtained within the energy relaxation time is obtained from the potential difference. Therefore, even if the potential gradient for supplying energy fluctuates greatly within the travel distance during the energy relaxation time, it is possible to obtain a physically valid energy distribution. Is high.
(実施例) 以下、本発明の詳細を図示の実施例によって説明す
る。(Examples) Hereinafter, details of the present invention will be described with reference to the illustrated examples.
第1図乃至第6図は本発明の一実施例を説明するため
のもので、この実施例は、電子伝導型の絶縁ゲート型ト
ランジスタにおける電子及び正孔の輸送現象をエネルギ
ーの保存式を含めて解析する例を示している。FIGS. 1 to 6 illustrate an embodiment of the present invention. In this embodiment, the electron and hole transport phenomena in an electron conduction type insulated gate transistor include an energy conservation equation. An example in which the analysis is performed is shown.
本実施例は、基本的には従来と同様に、半導体素子の
形状,半導体部分のドナー,アクセプタ分布,取り出し
電極の形状及び電位を読み取り、半導体素子形状の半導
体素子内の電位分布,電子濃度分布及び正孔濃度分布の
内の少なくとも一つの物理量を求め、且つ電子と正孔の
エネルギー分布の少なくとも一方を求めるために、前述
した5つの連立偏微分方程式(1)〜(5)を、数値的
に解き、半導体素子の電気的特性を評価する方法であ
る。従来方法と異なる点は、エネルギー保存式(4)
(5)中の単位時間当りのエネルギー供給項である電力
密度項を定める際、電子又は正孔がエネルギーを保持す
る平均時間であるエネルギー緩和時間の間に、電子又は
正孔が走行する始点と終点を検出し、該検出された始点
と終点の間の電位差をエネルギー緩和時間で割って終点
における電力密度項としたことにある。In the present embodiment, the shape of the semiconductor device, the donor and acceptor distribution of the semiconductor portion, the shape and potential of the extraction electrode, and the potential distribution and the electron concentration distribution in the semiconductor device having the semiconductor device shape are basically read as in the prior art. In order to determine at least one physical quantity of the hole concentration distribution and at least one of the electron and hole energy distributions, the above-described five simultaneous partial differential equations (1) to (5) are numerically calculated. This is a method for evaluating the electrical characteristics of a semiconductor device. The difference from the conventional method is the energy conservation formula (4)
(5) In defining the power density term, which is the energy supply term per unit time, the starting point at which electrons or holes travel during the energy relaxation time, which is the average time during which electrons or holes retain energy. The end point is detected, and the detected potential difference between the start point and the end point is divided by the energy relaxation time to obtain a power density term at the end point.
まず、第1図に示す電力項の計算手順を示すフローチ
ャートを用いて、本実施例による評価手順の概要を示
す。予め電位勾配の分布を求める(M1)と共に、電子及
び正孔の速度分布を求め(M2)、速度ベクトルに沿っ
て、エネルギー緩和時間の間に電子及び正孔が走行する
始点と終点を求める(M3)。そして、始点と終点の電位
差をエネルギー緩和時間で割って終点における電力とす
る(M4)。First, an outline of an evaluation procedure according to the present embodiment will be described using a flowchart showing a procedure for calculating a power term shown in FIG. The distribution of the potential gradient is determined in advance (M1), and the velocity distribution of electrons and holes is determined (M2), and the starting point and the ending point where the electrons and holes travel during the energy relaxation time are determined along the velocity vector ( M3). Then, the potential difference between the start point and the end point is divided by the energy relaxation time to obtain power at the end point (M4).
次に、第2図の等電位線分布と第3図の電子の速度ベ
クトル分布を用いて、電力項の計算方法を模式的に示
す。なお、図において、201,301はp型半導体基板、20
2,302はn+拡散層(ソース)、203,303はn+拡散層(ドレ
イン)、204,304は酸化膜、205,305はソース電極、206,
306はゲート電極、207,307はドレイン電極を示してい
る。電力を求めたい所望の点A、即ち走行の終点から、
第3図の速度ベクトルに沿って、エネルギー緩和時間τ
Wに相当する距離を遡った点Bを求める。遡る距離及び
方向は電子速度 を用いて、以下の積分を数値的に行うことによって求め
る。Next, a method for calculating a power term is schematically shown using the equipotential line distribution in FIG. 2 and the velocity vector distribution of electrons in FIG. In the figures, 201 and 301 are p-type semiconductor substrates, 20
2,302 is an n + diffusion layer (source), 203,303 are n + diffusion layers (drain), 204,304 are oxide films, 205,305 are source electrodes, 206,
306 denotes a gate electrode, and 207 and 307 denote drain electrodes. From a desired point A where power is to be obtained, that is, the end point of traveling,
According to the velocity vector of FIG. 3, the energy relaxation time τ
A point B which goes back a distance corresponding to W is obtained. Backward distance and direction are electron speed Is obtained by numerically performing the following integration using
そして、A,B間の電位差q△ψをポアソン方程式から
求め、q△ψ/τWをエネルギー保存式の電力項とす
る。正孔についても同様である。 Then, the potential difference q △ ψ between A and B is obtained from the Poisson equation, and q △ ψ / τ W is defined as the power term of the energy conservation equation. The same applies to holes.
なお、A点から速度ベクトルに沿って遡るのではな
く、次式 を用いて、反対に速度ベクトルが向いている方向にτW
に相当する距離を積分してもよい。その場合、求めた電
力は第2図中のB′点での値となる。Note that instead of going back along the velocity vector from point A, the following equation Τ W in the direction in which the velocity vector is pointing
May be integrated. In this case, the obtained power becomes a value at point B 'in FIG.
第4図に第2図中のC−C′で切った断面でのエネル
ギー分布を示す。実線が本実施例方法、破線が従来法に
よって求めたものである。従来法の方が大きなエネルギ
ー分布となっている。一般に、電子の散乱確率はエネル
ギーに対して単調増加関数であり、従来法では電流値を
過小評価することになる。第5図にドレイン電圧 に対するドレイン電流のID特性図を示す。白丸が実測値
で、実線が本実施例方法、破線が従来法による計算結果
である。このように、本実施例によれば実測をほぼ再現
できることから、物理的精度の高さが保証されることに
なる。FIG. 4 shows an energy distribution in a section cut along the line CC 'in FIG. The solid line is obtained by the method of the present embodiment, and the broken line is obtained by the conventional method. The conventional method has a larger energy distribution. In general, the electron scattering probability is a monotonically increasing function with respect to energy, and the conventional method underestimates the current value. Figure 5 shows the drain voltage 4 shows an ID characteristic diagram of a drain current with respect to FIG. The white circles are the measured values, the solid line is the calculation result by the method of the present embodiment, and the broken line is the calculation result by the conventional method. As described above, according to the present embodiment, since the actual measurement can be substantially reproduced, high physical accuracy is guaranteed.
第6図に本実施例における具体的なフローチャートの
一例、即ち半導体素子の電気的特性を評価する計算手順
を示す。まず、物理的考察に基づいて初期解を求め、ポ
アソン方程式(1)のみをガンメルの方法(H.K.Gumme
l,'A Self−consistent iterative scheme for one−di
mensional steady state transistor calculations'.IE
EE Trans,on ED.,ED−ll,p.455 1964年)で解き、電位
分布ψを更新する。ここで、電位分布が前回の電位分布
に比べて、例えば10mV程度以下の変化しかない場合は、
収束したと見做して次の処理に進む。FIG. 6 shows an example of a specific flowchart in this embodiment, that is, a calculation procedure for evaluating the electrical characteristics of the semiconductor element. First, an initial solution is obtained based on physical considerations, and only the Poisson equation (1) is calculated using the Gummel method (HKGumme
l, 'A Self-consistent iterative scheme for one-di
mensional steady state transistor calculations'.IE
EE Trans, on ED., ED-ll, p. 455 1964) and updates the potential distribution ψ. Here, if the potential distribution changes only about 10 mV or less compared to the previous potential distribution,
The process proceeds to the next process assuming that the convergence has been achieved.
次いで、電子及び正孔の電流連続方程式(2)(3)
を各々解いて、新たな電子分布n,正孔分布pを求める。
続いて、第1図の処理を用いて、電子速度に沿って、電
子エネルギー緩和時間に相当する距離を遡ることによ
り、電子の電力密度項を計算する。その後、前記電力密
度項を含めてエネルギー保存式(4)を解き、新たな電
子エネルギーwEを求める。Next, the current continuity equations for electrons and holes (2) and (3)
Are solved to obtain a new electron distribution n and a new hole distribution p.
Subsequently, by using the processing of FIG. 1, the power density term of the electrons is calculated by going back a distance corresponding to the electron energy relaxation time along the electron velocity. After that, the energy conservation equation (4) including the power density term is solved to obtain a new electron energy w E.
次いで、第1図の処理を用いて正孔速度に沿って、正
孔エネルギー緩和時間に相当する距離を遡ることによ
り、正孔の電力密度項を計算する。続いて、前記電力密
度項を含めてエネルギー保存式(5)を解き、新たな正
孔エネルギーwHを求める。そして、ポアソン方程式
(1)を解き、電位分布ψを修正する。Next, the power density term of the holes is calculated by going back the distance corresponding to the hole energy relaxation time along the hole velocity using the processing of FIG. Then, solving the energy conservation equation (5), including the power density section obtains a new hole energy w H. Then, the Poisson equation (1) is solved to correct the potential distribution ψ.
次いで、収束判定を行う。この判定は、例えばψ,n,
p,wE,wHの修正量の最大値が全て各々所定の値以下であ
るか否かを判定する。所定の値以上なら再び、電子の電
流連続式(2)を解く処理まで戻り、今回の解を初期値
として同じ処理を繰り返す。そして、収束したら必要に
応じてグラフィック・データ及び数値データを出力す
る。Next, convergence determination is performed. This determination is made, for example, by ψ, n,
It is determined whether or not all of the maximum values of the correction amounts of p, w E , and w H are each equal to or less than a predetermined value. If the value is equal to or more than the predetermined value, the process returns to the process of solving the electron current continuous equation (2), and the same process is repeated with the current solution as an initial value. Then, when it converges, it outputs graphic data and numerical data as needed.
このようにして各方程式(1)〜(5)を解くことに
より、半導体素子の電気特性を解析評価する。そしてこ
の場合、エネルギー緩和時間の間に電子や正孔が走行す
る距離の間で、エネルギーを供給する電位勾配が急峻に
変化する場合であっても、得られるエネルギー分布を過
大評価することはなく、妥当なエネルギーが得られる。
このため、エネルギー保存式の電力密度項を物理的な考
察から合理的に見積ることができ、特性予測の物理的精
度を高めることが可能となる。By solving the equations (1) to (5) in this manner, the electrical characteristics of the semiconductor element are analyzed and evaluated. And in this case, even if the potential gradient for supplying energy changes abruptly during the distance in which electrons and holes travel during the energy relaxation time, the obtained energy distribution is not overestimated. , Reasonable energy is obtained.
For this reason, the power density term of the energy conservation equation can be reasonably estimated from physical considerations, and the physical accuracy of characteristic prediction can be improved.
なお、本発明は上述した実施例に限定されるものでは
ない。実施例では、各方程式を別個に解くデカップル法
について示したが、一部或いは全ての方程式を一度に解
くカップル法を用いてもよい。また、電子エネルギー保
存式及び正孔エネルギー保存式として、(4)(5)式
の代わりに次の(4)′(5)′式を用いてもよい。The present invention is not limited to the embodiments described above. In the embodiment, the decoupling method for solving each equation separately has been described. However, a coupling method for solving some or all of the equations at once may be used. As the electron energy conservation equation and the hole energy conservation equation, the following equations (4) ′ (5) ′ may be used instead of the equations (4) and (5).
この場合、エネルギー保存式の電力密度項ではなく電
力項を定めることになる。その他、本発明の要旨を逸脱
しない範囲で、種々変形して実施することができる。 In this case, the power term is determined instead of the energy conservation type power density term. In addition, various modifications can be made without departing from the scope of the present invention.
[発明の効果] 以上詳述したように本発明によれば、エネルギー保存
式の電力項を見積る際、エネルギー緩和時間の間に走行
する始点と終点の間の電位差からエネルギー緩和時間内
に得られるエネルギーを求めることにより、非常に微細
な半導体素子内部で電子や正孔がエネルギー緩和時間に
走行する距離以内で、電位勾配が変動している場合で
も、正確なエネルギー分布を求めることができ、物理的
に精度の高い電気的特性の予測が可能となる。[Effects of the Invention] As described in detail above, according to the present invention, when estimating the power term of the energy conservation equation, it is obtained within the energy relaxation time from the potential difference between the start point and the end point traveling during the energy relaxation time. By obtaining energy, accurate energy distribution can be obtained even when the potential gradient fluctuates within the distance that electrons and holes travel within the energy relaxation time inside a very fine semiconductor device. This makes it possible to predict the electrical characteristics with high accuracy.
第1図乃至第6図はそれぞれは本発明の一実施例を説明
するためのもので、第1図は電力項の計算手順を示すフ
ローチャート、第2図は半導体素子内部の2次元断面図
中における等電位線分布を示す模式図、第3図は半導体
素子内部の2次元断面図中における電子速度分布を示す
模式、第4図は第1図中のC−C′の線で切った断面に
おける電子エネルギー分布を示す特性図、第5図は実施
例と従来法による計算結果を実測値と比較して示す特性
図、第6図は半導体素子の電気的特性を評価する計算手
順を示すフローチャート、第7図乃至第10図はそれぞれ
従来方法を説明するためのもので、第7図はデバイス・
シミュレーションの離散化用格子点を示す模式図、第8
図は電力項の計算手順を示すフローチャート、第9図は
電力項の計算の物理的意味を示す概念図、第10図は電力
項の計算の物理的限界を示す概念図である。 201,301……P型半導体基板、 202,302……n+拡散層(ソース)、 203,303……n+拡散層(ドレイン)、 204,304……酸化膜、 205,305……ソース電極、 206,306……ゲート電極、 207,307……ドレイン電極。1 to 6 are diagrams for explaining one embodiment of the present invention. FIG. 1 is a flowchart showing a procedure for calculating a power term, and FIG. 2 is a two-dimensional sectional view showing the inside of a semiconductor device. FIG. 3 is a schematic diagram showing an electron velocity distribution in a two-dimensional cross-sectional view of the inside of a semiconductor device, and FIG. 4 is a cross-sectional view taken along a line CC ′ in FIG. FIG. 5 is a characteristic diagram showing the comparison between the results of the calculation according to the embodiment and the conventional method and the measured values, and FIG. 6 is a flowchart showing a calculation procedure for evaluating the electrical characteristics of the semiconductor device. 7 to 10 are for explaining the conventional method, and FIG.
Schematic diagram showing grid points for discretization of simulation, FIG.
FIG. 9 is a flowchart showing the procedure for calculating the power term, FIG. 9 is a conceptual diagram showing the physical meaning of the calculation of the power term, and FIG. 10 is a conceptual diagram showing the physical limit of the calculation of the power term. 201,301 ... P-type semiconductor substrate, 202,302 ... n + diffusion layer (source), 203,303 ... n + diffusion layer (drain), 204, 304 ... oxide film, 205,305 ... source electrode, 206,306 ... gate electrode, 207,307 ... ... Drain electrode.
Claims (3)
び正孔濃度分布の内の少なくとも一つの物理量を求め、
且つ電子と正孔のエネルギー分布の少なくとも一方を求
めるために、ポアソン方程式,電子電流連続式,正孔電
流連続式,電子エネルギー保存式及び正孔エネルギー保
存式を解くことによって、半導体素子の電気的特性を評
価する方法において、 前記エネルギー保存式中の単位時間当りのエネルギー供
給項である電力項又は電力密度項を定める際、電子又は
正孔がエネルギーを保持する平均時間であるエネルギー
緩和時間の間に、電子又は正孔が走行する始点と終点を
検出し、該検出された始点と終点の間の電位差をエネル
ギー緩和時間で割って終点における電力項又は電力密度
項としたことを特徴とする半導体素子の特性評価方法。And determining at least one physical quantity among a potential distribution, an electron concentration distribution, and a hole concentration distribution in the semiconductor element.
In addition, in order to determine at least one of the electron and hole energy distributions, the Poisson equation, the electron current continuation formula, the hole current continuation formula, the electron energy conservation formula, and the hole energy conservation formula are solved. In the method of evaluating characteristics, when determining a power term or a power density term which is an energy supply term per unit time in the energy conservation equation, during an energy relaxation time which is an average time during which electrons or holes retain energy. A semiconductor wherein a starting point and an ending point where electrons or holes travel are detected, and a potential difference between the detected starting point and the ending point is divided by an energy relaxation time to obtain a power term or a power density term at the ending point. Method for evaluating device characteristics.
子又は正孔のエネルギー緩和時間をτW間の走行の終点
を指定した後に、その終点を積分の起点として、 によって、走行の始点を求めることを特徴とする請求項
1記載の半導体素子の特性評価方法。Wherein the electrons or holes of the velocity vector is V, after the electrons or holes energy relaxation time specifies the end point of the travel between tau W, the end point as a starting point of the integration, 2. The method for evaluating characteristics of a semiconductor device according to claim 1, wherein the starting point of the traveling is determined by the following method.
子又は正孔のエネルギー緩和時間をτWとして、τWと
し、τW間の走行の始点を指定した後に、その始点を積
分の起点として、 によって、走行の終点を求めることを特徴とする請求項
1記載の半導体素子の特性評価方法。3. The electron or hole velocity vector is V, the electron or hole energy relaxation time is τ W , τ W, and the starting point of travel between τ W is specified. As a starting point, 2. The method for evaluating characteristics of a semiconductor device according to claim 1, wherein an end point of the traveling is obtained by the following.
Priority Applications (1)
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JP23008790A JP3247367B2 (en) | 1990-08-31 | 1990-08-31 | Method for evaluating characteristics of semiconductor device |
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JP23008790A JP3247367B2 (en) | 1990-08-31 | 1990-08-31 | Method for evaluating characteristics of semiconductor device |
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JPH04111439A JPH04111439A (en) | 1992-04-13 |
JP3247367B2 true JP3247367B2 (en) | 2002-01-15 |
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