JP3579332B2 - Method and apparatus for predicting damage to wiring structure having protective film on surface - Google Patents

Description

ここで、Ｎ：原子濃度（密度）、Ｄ_０：振動数項、ｋ：ボルツマン定数、Ｔ：絶対温度、Ｑ：活性化エネルギ、Ｚ^＊：有効電荷数、ｅ：単位電荷，ρ：電気抵抗率、ｊ：電流密度ベクトル、σ_Ｔ：引張りの熱応力、Ω：原子体積、Ｎ_０：無応力状態における原子濃度（密度）、Ｎ_Ｔ：σ_Ｔが作用したときの原子濃度（密度）、κ：体積弾性率である。
【０００７】
（多結晶配線）
多結晶配線内における原子拡散径路として、結晶粒界と結晶粒内が考えられるが、結晶粒内における原子の易動度は粒界におけるそれと比較し無視できるほど小さいので、結晶粒界における発散のみを考える。多結晶配線における結晶粒界を考慮するために、結晶粒界構造を平均化したモデルを導入する。図１は多結晶配線構造のモデルの例である。図１に示すように、平均結晶粒径ｄの（√３）／６倍の長さを持つ三本の結晶粒界で構成される三重点を、内部に一つだけ含む単位厚さの四角形要素を仮定する。同要素の面積は（√３）ｄ^２／４である。結晶粒界ＩＩおよびＩＩＩは結晶粒界Ｉに対して対称でありその狭角は１２０°に近いが、わずかな偏差２Δψが存在するものとする。
【０００８】
電流密度ベクトルｊのｘ方向成分およびｙ方向成分をｊ_ｘ，ｊ_ｙ、原子濃度をＮ、原子濃度勾配のｘ方向成分およびｙ方向成分をそれぞれ∂Ｎ／∂ｘ，∂Ｎ／∂ｙとし、結晶粒界Ｉとｘ軸のなす角をθとすると、結晶粒界Ｉ，ＩＩおよびＩＩＩの端部における電流密度、温度、原子濃度、原子濃度勾配は、図１に示すように表される。これらを式（１）に代入することにより、それぞれの粒界端における結晶粒界に沿った原子流束が得られる。ここに要素から外に出る方向を正と定義する。結晶粒界Ｉ，ＩＩおよびＩＩＩの端部における原子流束に粒界の幅δおよび単位厚さを乗じることにより、それぞれの粒界端における単位時間あたりの原子の移動数を得、それらを各々加える。微小項を無視し、さらに電流保存則を用いて式を簡単化した後、要素の体積（√３）ｄ^２／４で除す。このようにして、結晶粒界Ｉとｘ軸のなす角がθなる場合の結晶粒界拡散における単位時間、単位体積あたりの原子の減少数、すなわち原子流束の発散ＡＦＤ_{ｇｂ θ} ^Ｒは次のように与えられる。
【数２】

ここに、Ｃ_ｇｂ ^Ｒ＝ＮδＤ_０／ｋである。ＡＦＤ_{ｇｂ θ} ^Ｒは原子濃度勾配の影響を受けたＥＭによる単位時間、単位体積あたりの原子の減少数すなわち原子流束の発散を表す。式（２）の右辺における＜＞内の第一項および第ニ項は結晶粒界三重点での原子流束の発散に関係する項、その他の項は結晶粒界自身での原子流束の発散に関係した項である。また、ＡＦＤ_{ｇｂ θ} ^Ｒが正の値をとる場合はボイド（空隙）の形成に、負の値をとる場合はヒロック（表面の突起）の形成に寄与する。実際の配線を考えた場合、θは任意の値をとる。よってθのとり得る全ての範囲（０＜θ＜２π）を考慮した流束の発散を求める必要がある。ここでボイド形成のみに着目するものとして、θが０から２πまで変化する場合のＡＦＤ_{ｇｂ θ} ^Ｒの正値のみの期待値を求める。ここに、ＡＦＤ_{ｇｂ θ} ^Ｒの負の値は、ボイド形成に寄与しないため、０とみなす。ＡＦＤ_{ｇｂ θ} ^ＲとＡＦＤ_{ｇｂ θ} ^Ｒの絶対値の和をとり、２で除すことによって、多結晶配線の結晶粒界におけるボイド形成に関する原子流束の発散ＡＦＤ_ｇｅｎ ^Ｒを次式のように導出する。
【数３】

【０００９】
（バンブー配線）
一方、バンブー配線においては、結晶粒界での原子流束の発散は無視することができることから、結晶粒内での発散のみを考慮すると、単位時間、単位体積あたりの原子の減少数、すなわち原子流束の発散ＡＦＤ_ｌａｔ ^Ｒは次式で表される。
【数４】

なお、ここでは、電流保存則を用いるとともに、微小項を無視して簡単化を行っている。仮にボイド形成に寄与するＡＦＤ_ｌａｔ ^Ｒの正値のみを抽出するとすれば、ＡＦＤ_ｇｅｎ ^Ｒは次式で与えられる。
【数５】

【００１０】
＜物性値の導出法＞
（多結晶配線）
保護膜の効果を考慮した多結晶配線におけるＥＭ損傷支配パラメーターＡＦＤ_ｇｅｎ ^Ｒに含まれる薄膜の物性に関する定数は、平均結晶粒径ｄ，結晶粒界間の相対角に関する定数Δψ，結晶粒界拡散における活性化エネルギＱ_ｇｂ，有効電荷数Ｚ^＊，Ｃ_ｇｂ ^Ｒ＝ＮδＤ_０／ｋである。電流密度および温度が一定とみなせ、扱いの容易な直線状配線中央部を対象に、これらの定数の分離について以下に示す。
まず、直線状配線中央部ではＮ≒Ｎ_Ｔとおける。ここにＣ_ｇｂ ^Ｒ＝ＮδＤ_０／ｋは本来Ｎに関する変数であるが、配線のとり得る応力値を考えると、ＮのＮ_０からの変動は高々数％であると考えられるので、近似的にＮ＝Ｎ_Ｔ≒Ｎ_０とおけば、Ｃ_ｇｂ ^Ｒ＝Ｎ_０δＤ_０／ｋとなり薄膜に関する物性値とみなせる。
また、濃度勾配∂Ｎ／∂ｘは配線長さに反比例するもの（Ｂｌｅｃｈ，Ｉ．Ａ．， Ｊ．Ａｐｐｌ．Ｐｈｙｓ．，４７−４（１９７６），１２０３参照）であり、配線形状に依存する値である。ここでは配線中央部で濃度分布を線形とみなし、κ・∂Ｎ／∂ｘを配線形状に依存した定数として導出する。Δψについては、同じ条件で同時に作製した表面が露出した配線より導出した値を用いることとする。なお、物性値の導出に必要な配線の電流密度、電流密度勾配、温度および温度勾配の分布を求めるために、実験に用いた配線を対象とした有限要素解析等の数値解析を実施する。
【００１１】
まず、平均結晶粒径ｄは、集束イオンビーム（ＦＩＢ）装置等により計測する。また活性化エネルギＱ_ｇｂ ^＊［＝Ｑ_ｇｂ−σ_ＴΩ］は、次のように決定する。同一の入力電流密度ｊ_ｌに対して、三種類の異なる基板温度Ｔ_ｓ１，Ｔ_ｓ２およびＴ_ｓ３の下で一定時間通電するＥＭ加速実験を行う。ここで基板温度Ｔ_ｓ１のときの配線中央部の温度をＴ_ｌ，Ｔ_ｓ２のときをＴ_２およびＴ_ｓ３のときをＴ_３とし、各実験条件での配線中央部における条件を条件１：ｊ_１およびＴ_１，条件２：ｊ_１およびＴ_２，条件３：ｊ_１およびＴ_３とする。これに加え、ｊ_１と異なる入力電流密度ｊ_４に対して基板温度を調節することにより、配線中央部の温度をＴ_２と等しくした条件４：ｊ_４およびＴ_２でも一定時間通電するＥＭ加速実験を行う。通電後、それぞれについて、配線中央部付近のボイド体積を計測する。温度勾配、電流密度勾配および濃度勾配の傾きに関する項を無視した直線形状配線中央部におけるＡＦＤ_ｇｅｎ ^Ｒを、条件１，２および３に対して表示する。それぞれに、ボイド体積測定領域の面積、配線の厚さ、通電時間から潜伏期間を除いた正味の通電時間および原子体積を乗じる。これらと、実際に計測したボイド体積は等しいことより、例えば、条件１における配線中央部のボイド計測値をＶ_１とすると、この場合次の式が成り立つ。
【数６】

ここに
【数７】

【数８】

Ａはボイド体積測定領域の面積，Ｑ_ｇｂ ^＊＝Ｑ_ｇｂ−σ_ＴΩ，ρ_１は温度Ｔ_１における配線の抵抗率，ｔ_１は条件１における通電時間から潜伏期間を除いた正味の通電時間，ｔｈｉｃｋは配線の厚さである。式（６）と同様な方程式が条件２および３に対しても成り立つ。式（６）において両辺自然対数をとり、条件１における式から条件３における式を減じることにより、Ｑ_ｇｂ ^＊と有効電荷数Ｚ^＊および原子濃度勾配κ・∂Ｎ／∂ｘの関係式（９）が得られる。
【数９】

ここに
【数１０】

ρ_３は温度Ｔ_３における配線の抵抗率，ｔ_３は条件３における通電時間から潜伏期間を引いた正味の通電時間である。なおここでは、Ｔ_１とＴ_３間のσ_Ｔの変動は微小とみなして無視した。式（９）と同様な式が条件１と２および条件２と３においても成り立つ。一方、配線中央部の温度が同じで電流密度がｊ_１，ｊ_４と異なる条件２と４に関して式（６）と同様な式を減じることにより、Ｚ^＊とκ・∂Ｎ／∂ｘの関係式が求まる。
【数１１】

ここでρ_２は温度Ｔ_２における配線の抵抗率、Ｖ_２およびＶ_４はそれぞれ条件２および条件４におけるボイド体積測定値、ｔ_２およびｔ_４は条件２および条件４における通電時間から潜伏期間を引いた正味の通電時間である。式（１１）を式（９）に代入すると、条件１と３におけるＱ_ｇｂ ^＊（Ｑ_ｇｂ１３ ^＊と表す）とκ・∂Ｎ／∂ｘだけの関係式が式（１２）のように得られる。なお、ＥＭ損傷のしきい電流密度が実験的に求めることが可能な場合には、条件４は必要とせず後述の式（１４）と同様なＺ^＊とκ・∂Ｎ／∂ｘの関係式を得ることができるので、式（１１）のかわりに用いることができる。
【数１２】

式（１２）と同様な式が、条件１と２および条件２と３の間においても成り立つので、Ｑ_ｇｂ ^＊とκ・∂Ｎ／∂ｘだけの関係式が３式得られることになる。ここでＱ_ｇｂ ^＊は各式で同じ値をとり得るはずだから、三式間のＱ_ｇｂ ^＊の偏差（Ｑ_ｇｂ１３ ^＊−Ｑ_ｇｂ２３ ^＊）^２＋（Ｑ_ｇｂ２３ ^＊−Ｑ_ｇｂ１２ ^＊）^２＋（Ｑ_ｇｂ１２ ^＊−Ｑ_ｇｂ１３ ^＊）^２が最も小さくなるように、適当なκ・∂Ｎ／∂ｘを定めることにより、Ｑ_ｇｂ ^＊およびκ・∂Ｎ／∂ｘが求まる。次いでＺ^＊は得られたκ・∂Ｎ／∂ｘを式（１１）に代入することにより求まる。さらに、定数Ｃ_ｇｂ ^Ｒは条件１，２および３に対して、式（６）における左辺のボイド体積測定値と、右辺のＡＦＤ_ｇｅｎ ^Ｒを用いた数値計算によるボイド体積が一致するように決定する。これより条件１，２および３に対してそれぞれのＣ_ｇｂ ^Ｒの値が得られるが、これらを平均したものをＣ_ｇｂ ^Ｒの値とする。
また、配線内部の熱応力を見積る場合には、上記方法で得たＱ_ｇｂ ^＊［＝Ｑ_ｇｂ−σ_ＴΩ］と保護膜のない配線を用いて得たＱ_ｇｂの値からσ_Ｔを見積ることができる。
以上のようにすべての物性値は直線形状の配線を用いた簡単な実験から求まる。
【００１２】
（バンブー配線）
保護膜の効果を考慮したバンブー配線におけるＥＭ損傷支配パラメータＡＦＤ_ｇｅｎ ^Ｒに含まれる薄膜の物性に関する定数は、Ｑ_ｌａｔ，Ｚ^＊，Ｄ_０である。直線形状のバンブー配線においてボイド形成領域である同配線陰極部を対象としたこれらの定数の導出法について示す。
まず式（４）中において直線状配線陰極部ではＮ≒Ｎ_Ｔとおける。ここでも配線陰極部で濃度分布を線形とみなし、κ・∂Ｎ／∂ｘを配線形状に依存した定数として導出する。
活性化エネルギＱ_ｌａｔ ^＊［＝Ｑ_ｌａｔ−σ_ＴΩ］は、次のように決定する。同一の入力電流密度ｊ_ｌに対して三種類の異なる基板温度Ｔ_ｓ１，Ｔ_ｓ２およびＴ_ｓ３の下で一定時間通電するＥＭ加速実験を行う。ここで基板温度Ｔ_ｓ１のときの配線陰極部の温度をＴ_１，Ｔ_ｓ２のときをＴ_２および_Ｔｓ３のときをＴ_３とし、各実験条件での配線陰極部における条件を条件１：ｊ_ｌおよびＴ_１，条件２：ｊ_１およびＴ_２，条件３：ｊ_ｌおよびＴ_３とする。通電後それぞれについて、配線陰極部のボイド体積を計測する。∂^２Ｎ／∂ｘ^２＝０と仮定した直線形状配線陰極部におけるＡＦＤ_ｇｅｎ ^Ｒを、条件１，２および３に対して表示し、それぞれにボイド体積測定領域の面積、配線の厚さ、通電時間から潜伏期間を除いた正味の通電時間および原子体積を乗じる。これらと実際に計測したボイド体積を等値することより、例えば条件１における配線陰極部のボイド体積計測値をＶ_１とすれば、次式が成り立つ。
【数１３】

ここに、Ａはボイド体積測定領域の面積，ρ_１は温度Ｔ_１における配線の抵抗率，ｔ_１は条件１における正味の通電時間，ｔｈｉｃｋは配線の厚さである。式（１３）と同様な方程式が条件２および３に対しても成り立つ。式（１３）において両辺自然対数をとり、条件１における式から条件３における式を減じることにより、Ｑ_ｌａｔ ^＊有効電荷数Ｚ^＊および原子濃度勾配κ・∂Ｎ／∂ｘの関係式が得られる。ところで、電流密度が小さい場合、ＥＭによる原子流束と濃度勾配による逆拡散はちょうど釣り合い、みかけの原子流束が零となるためＥＭ損傷が生じないが、ある電流密度を超えるとＥＭによる原子流束が濃度勾配による逆拡散に勝り、ＥＭ損傷が生じる。このときのしきい電流密度をｊ_ｔｈとする。陰極部において式（６）が零となるとき、次式のようなＺ^＊とκ・∂Ｎ／∂ｘの関係が成り立つ。
【数１４】

ここにρ_４はｊ_ｔｈ作用下の陰極部の温度における抵抗率であり、ｊ_ｔｈは実験的に求まる。式（１４）をＱ_ｌａｔ ^＊と有効電荷数Ｚ^＊および原子濃度勾配κ・∂Ｎ／∂ｘの関係式に代入することにより、κ・∂Ｎ／∂ｘとＱ_ｌａｔ ^＊の関係を次式のように得る。
【数１５】

ここに条件１と３で定まるＱ_ｌａｔ ^＊をＱ_{ｌａｔ１３} ^＊とおいた。また、ｔ_３は条件３における正味の通電時間である。式（１５）と同様な式が、条件１と２および条件２と３の間においても成り立つので、Ｑ_ｌａｔ ^＊とκ・∂Ｎ／∂ｘだけの関係式が３式得られることになる。ここでκ・∂Ｎ／∂ｘは各式で同じ値をとるはずであるから、３式間のκ・∂Ｎ／∂ｘの偏差（κ・∂Ｎ／∂ｘ_２３−κ・∂Ｎ／∂ｘ_１３）^２＋（κ・∂Ｎ／∂ｘ_２３−κ・∂Ｎ／∂ｘ_１２）^２＋（κ・∂Ｎ／∂ｘ_１２一κ・∂Ｎ／∂ｘ_１３）^２が最も小さくなるように、適当なＱ_ｌａｔ ^＊を定めることにより、Ｑ_ｌａｔ ^＊およびκ・∂Ｎ／∂ｘが求まる。ここに、条件１と３で定まるκ・∂Ｎ／∂ｘをκ・∂Ｎ／∂ｘ_１３、条件２と３で定まるκ・∂Ｎ／∂ｘをκ・∂Ｎ／∂ｘ_２３、条件１と２で定まるκ・∂Ｎ／∂ｘをκ・∂Ｎ／∂ｘ_１２とおいた。
次いで、Ｚ^＊は得られたκ・∂Ｎ／∂ｘを式（１４）に代入することにより求まる。さらに、定数Ｄ_０はＮ_Ｔ＝Ｎ_０とし、式（１３）における左辺のボイド体積計測値と、右辺のＡＦＤ_ｇｅｎ ^Ｒを用いた数値計算によるボイド体積が、一致するように決定する。
また、配線内部の熱応力を見積る場合には、上記方法で得たＱ_ｌａｔ ^＊［＝Ｑ_ｌａｔ−σ_ＴΩ］と保護膜のない配線を用いて得たＱ_ｌａｔ値からσ_Ｔを見積ることができる。
以上のように、すべての物性定数は直線形状の配線を用いた簡単な実験から求まる。
【００１３】
＜断線過程のシミュレーション＞
本発明におけるシミュレーションは、予測対象の配線を要素分割し、各々の要素厚さをＡＦＤ_ｇｅｎ ^Ｒに基づき減少させることにより、断線過程をシミュレーションするものである。ここで、多結晶配線の場合は、結晶粒界に沿ってスリット状に成長するというボイドの成長形態を考慮し、実験・観察より得た配線の平均結晶粒径および平均スリット幅に基づき配置した、スリット状要素のみで要素厚が減少するものとする。一方、バンブー配線の場合にはスリット状要素の設置は必要ない。図２に多結晶配線１００の要素分割の一例を示す。
多結晶配線の場合を例にとって、断線過程のシミュレーション処理を図３に示すフローチャートで説明する。まず、配線の電流密度分布および温度分布を、有限要素解析等の数値解析により求める（Ｓ２０４）。各要素のＡＦＤ_ｇｅｎ ^Ｒの値は、これらの分布と予め加速試験から決定した薄膜の物性値を用いて計算する（Ｓ２０６）。各要素のＡＦＤ_ｇｅｎ ^Ｒにの計算に用いる原子濃度の初期値は、このルーティンの最初に与えられている（Ｓ２０２）。この際、ボイド形成に寄与するＡＦＤ_{ｇｂ θ} ^Ｒの正値に着目したＡＦＤ_ｇｅｎ ^Ｒ値と、ヒロック形成に寄与するＡＦＤ_{ｇｂ θ} ^Ｒの負値のみに着目したＡＦＤ_ｇｅｎ ^Ｒ値の双方について計算する。シミュレーションの１ステップにおける各要素の金属原子数の変化量は、要素の体積および１ステップの時間を各々のＡＦＤ_ｇｅｎ ^Ｒに乗じることによりボイド形成寄与分とヒロック形成寄与分に分けて求める（Ｓ２０８）。さらに、これを用いて各要素における濃度初期値Ｎ_０からの濃度の変化量をボイド及びヒロック形成寄与分に分けて求める（Ｓ２１０）。次に、これらを加えることにより各要素の濃度Ｎを求め（Ｓ２１４）、配線全体の濃度分布を得る。ここで、濃度の値がボイドの形成に至る臨界値Ｎ_ｍｉｎを下まわるか、ヒロックの形成に至る臨界値Ｎ_ｍａｘを上まわるかの判定を行い（Ｓ２１６）、この範囲内の場合（Ｓ２１６でＹｅｓ）は、ＥＭ損傷の潜伏期間として、得られた原子濃度を用いて再度ＡＦＤ_ｇｅｎ ^Ｒ値の計算を行い、以降の計算を繰り返す。一方、シミュレーションの進行に伴い、濃度がＮ_ｍｉｎからＮ_ｍａｘの範囲を超えた場合は（Ｓ２１６でＮｏ）、ＥＭ損傷の進行期間として、ボイド形成に寄与する原子濃度変化量より算出した原子の減少量に応じ、スリット状要素の厚さを減少させる（Ｓ２２０）。その際、スリット状要素とその両隣の要素の濃度変化量を用いて、スリット状要素の厚さを減少させるが、バンブー配線の場合にはこの処理は必要ない。
【００１４】
ここで、Ｎ_ｍｉｎ，Ｎ_ｍａｘの値およびκ値は、加速試験の結果から次のようにあらかじめ決定しておく。まず、直線形状配線を用いた加速試験により、ある条件下の潜伏期間および配線中央部でのκ・∂Ｎ／∂ｘの値を得る。次に同様な条件下で先のシミュレーションの潜伏期間に関する部分のみ行い、加速試験で得た潜伏時間後にκ・∂Ｎ／∂ｘが等しくなるようにκの値を決定する。さらに、その時点での濃度Ｎの最大値をＮ_ｍａｘ値，最小値をＮ_ｍｉｎ値とする。なお、予め数値計算や実験計測などによりκ値が既知の場合には、このκ値を用いて試験で得た潜伏期間分のシミュレーションを行い、その時点での濃度Ｎの最大値をＮ_ｍａｘ値，最小値をＮ_ｍｉｎ値とする。逆にＮ_ｍａｘ，Ｎ_ｍｉｎが既知の場合には、潜伏期間のシミュレーションを行い、潜伏期間後のＮ_ｍａｘあるいはＮ_ｍｉｎが既知の値と一致するようにκ値を決定することができる。
厚さの減少した要素においては、厚さの減少量に対応した深さのボイドが形成されたものとみなす（Ｓ２２０）。各要素の厚さを考慮して（Ｓ２２６）、再度、電流，温度の有限要素解析等の数値解析を行う。
以上の計算を繰返し、融点を超えた温度の要素または、厚さがしきい値以下の要素が配線幅にわたって繋がった状態を断線と定義し（Ｓ２２２）、シミュレーションを終了する（Ｓ２２４）
【００１５】
＜短絡故障のシミュレーション＞
短絡故障のシミュレーションは、前に説明した図３のフローチャートの処理で潜伏期間のみの計算を行うものである。即ち、ステップＳ２１６において、濃度ＮがＮ_ｍａｘを上まわった時点で保護膜の破壊が生じたものとして計算を終了する。
Ｎ_ｍａｘ値は前節と同様な方法で与えることができるが、加速試験の保護膜と予測対象のそれが異なる場合や予測精度を上げる場合には、数値計算や実験計測などにより、κ値および保護膜の破壊強度あるいは破壊靱性を求め、そのときの応力値とκ値からＮ_ｍａｘの値を決定して与える。保護膜の破壊は短絡故障の以前に生じるものであり厳密には異なるが、第一近似的に保護膜の破壊時点を短絡故障の時点とした。
【００１６】
本発明の保護膜を有する金属配線のＥＭ損傷予測を行うプログラムを格納した記憶媒体から、プログラムをシステムで読み出して実行することにより、本発明の構成を実現することができる。この記録媒体には、フロッピー・ディスク、ＣＤ−ＲＯＭ、磁気テープ、ＲＯＭカセット等がある。
上述した本発明の保護膜を有する金属配線のＥＭ損傷予測を行うために用いるシステムとしては、スタンド・アローンのコンピュータ・システムばかりではなく、複数のシステムから構成される例えばクライアント・サーバ・システム等を用いてもよい。
【００１７】
【実施例】
以下に、上述の物性値の導出および断線等の予測の具体的な実施例として、以下に説明する。
＜多結晶配線における支配パラメータに含まれる物性値の導出＞
図４は、上述の考え方で物性定数を実際に求めたＡｌ薄膜配線１００の寸法形状を示す。図４に示すように、表面に酸化膜を形成したＳｉ基板上に、Ａｌ薄膜配線を形成後、その表面にポリイミド樹脂膜を被覆したサンプルを作製した。この配線に対して上述の物性値の導出法に従ってＥＭ加速試験を行った。作成した配線は２種類で、ＳａｍｐｌｅＬ：幅（Ｗ）８．６μｍ、長さ（Ｌ）８２μｍと、ＳａｍｐｌｅＳ：幅（Ｗ）８．７μｍ、長さ（Ｌ）３２μｍである。
ＥＭ加速試験は、図５に示すような構成を有する装置３００を用いて行った。活性化エネルギ導出のため、基板３６０の温度を、ホットステージ３５０を用いて、４５８，４７３，４８５Ｋの三種類の温度のいずれかに保持している。配線部１００における電流密度が約５．５ＭＡ／ｃｍ^２になるような定電流を、定電流源３１０から一定時間入出力させることに加え、配線中央部の温度が基板温度４７３Ｋのときの配線中央部の温度と同じくなるように、基板温度を４１５Ｋに保持して、配線部における電流密度がＳａｍｐｌｅＬで８．５ＭＡ／ｃｍ^２，ＳａｍｐｌｅＳで９．０ＭＡ／ｃｍ^２になるような定電流を、プローブ３７２および３７４から一定時間入出力した。いずれの条件に対しても２５本の試験片を用いた。全ての通電終了後、保護膜を除去し、電界放出型電子顕微鏡（ＦＥ−ＳＥＭ）４００により配線の中央部領域を観察した。得られたＳＥＭ像を画像解析することにより、ボイドの総面積を計測し、ボイドの面積に膜厚を乗じることにより、配線中央部におけるボイド体積を推定した。
【００１８】
保護膜の効果を考慮したＥＭ損傷支配パラメータＡＦＤ_ｇｅｎ ^Ｒを表面に保護膜を有する配線に適用し、ＳａｍｐｌｅＬおよびＳａｍｐｌｅＳのそれぞれに対してＡＦＤ_ｇｅｎ ^Ｒに含まれる物性値導出を行った。得られた物性値Ｑ_ｇｂ ^＊，Ｚ^＊，Ｃ_ｇｂ ^Ｒならびにκ・∂Ｎ／∂ｘの値を表１にまとめて示す。
【表１】
表１ ＡＦＤ_ｇｅｎ ^Ｒに含まれる物性値

表１から分かるように、ＳａｍｐｌｅＬとＳａｍｐｌｅＳの形状の違いによらず、Ｑ_ｇｂ ^＊，Ｚ^＊，Ｃ_ｇｂ ^Ｒの各値はほぼ同様であった。つまり、いずれも形状によらない物性値として機能しているといえる。また、配線内の応力（濃度）勾配は配線長さに反比例すると言われているが、表１において、κ・∂Ｎ／∂ｘの値は反比例はしていないものの、配線長さが短い方が１．２倍程度大きくなっている。Ｑ_ｇｂ ^＊およびＺ^＊について得られた値は、Ａｌ薄膜におけるこれまで報告されている値の間にあり妥当であるといえる。
以上から、表面に保護膜を有する配線にＡＦＤ_ｇｅｎ ^Ｒを適用した場合、保護膜の効果を的確に捉え、配線形状に依存しない物性値および形状に依存する濃度勾配を的確に導出することができた。
【００１９】
一方、ＡＦＤ_ｇｅｎ ^Ｒを保護膜を有する配線と同形状の、表面に保護膜が無い配線に適用した場合、得られた活性化エネルギＱ_ｇｂ ^＊（＝Ｑ_ｇｂ）はＳａｍｐｌｅＬと同形状の配線（以下ＳａｍｐｌｅＬｌと示す）で０．５４７ｅｖ、ＳａｍｐｌｅＳと同形状の配線（以下ＳａｍｐｌｅＳｌと示す）で０．５４９ｅｖであった。保護膜を有する配線の活性化エネルギと保護膜がない配線の活性化エネルギを比較すると、保護膜を有する配線のほうがＳａｍｐｌｅＬ，ＳａｍｐｌｅＳ共に小さくなっている。もし、界面拡散の影響が大きいとすると、活性化エネルギは大きくなるはずであるが、活性化エネルギは保護膜を有する配線で小さくなっているので、保護膜被覆に伴う界面拡散のＥＭ損傷への影響は小さいと考えられる。
また、Ｑ^＊ _ｇｂ＝Ｑ_ｇｂ−σ_ＴΩより、ＳａｍｐｌｅＬおよびＳａｍｐｌｅＳのそれぞれで熱応力を見積ったところ、それぞれから得た熱応力は、ＳａｍｐｌｅＬで２９ＭＰａであり、ＳａｍｐｌｅＳで４１５ＭＰａであった。
【００２０】
（断線予測）
図４に示す直線形状のＡｌ配線を断線予測の対象として想定する。試験片の物性値は、作製した試験片を用いた加速試験により得られた、表１の値を用いた。ただし、ここではκは一般的に知られている値を用いた。
断線予測においては、図４に示す、ＳａｍｐｌｅＬの値を持つ配線形状で、入力電流密度：５．５ＭＡ／ｃｍ^２、基板温度：４７３Ｋの動作条件で使用した場合を想定した。図６に断線予測の結果を図示する。図６に示すように、陰極側に損傷が起こることを示している。本シミュレーションによる予測では、配線陰極部付近における３５４１５ｓｅｃでの断線を予測した。
以上のように、保護膜を有する配線の寿命、および断線箇所についても予測が可能であった。
【００２１】
【発明の効果】
上述した本発明により、実際の配線構造により即した、保護膜を有する配線を対象とした高精度な断線（断線箇所，寿命）予測および配線を被覆する保護膜の破損（短絡箇所，寿命）予測が可能となる。また、保護膜を有する配線のためのＥＭ損傷支配パラメータに含まれる物性定数の決定法を用いることにより、配線内部の熱応力の評価および保護膜と配線の相互作用による拡散係数（活性化エネルギ）の評価が可能となり、集積回路を設計する上での保護膜及び配線材料の選択や強度評価の指標を与えることが可能となる。
【図面の簡単な説明】
【図１】多結晶配線構造のモデルの例を示す図である。
【図２】多結晶配線の要素分割の例を示す図である。
【図３】多結晶配線の場合の断線過程のシミュレーション処理を示すフローチャートである。
【図４】物性定数を求めた寸法形状を示す図である。
【図５】ＥＭ加速試験を行う装置構成を示す図である。
【図６】断線予測の結果を図示した図である。[0001]
TECHNICAL FIELD OF THE INVENTION
The present invention relates to prediction of the life and the like of metal wiring such as a semiconductor integrated circuit and a printed circuit board, and more particularly to a technique of predicting the life and the like of a wiring covered with a protective film.
[0002]
[Prior art]
The life expectancy of wiring of a semiconductor integrated circuit or the like has been performed using an empirical formula. In using this empirical formula, it is necessary to determine a constant depending on the wiring shape in the formula, so that universal prediction cannot be performed, which is complicated. Also, depending on the setting of the experimental conditions of the energization experiment performed to determine the constants in the equations, the prediction results differed, and the accuracy was not generally high.
Recently, the dominant parameter of electromigration damage (EM damage), which is a main factor of disconnection failure, has been theoretically formulated by the inventors, and a highly accurate and universal disconnection prediction method using this has been developed. . However, this method is intended for wiring without a protective film on the surface.
Practical wiring is generally covered with a protective film. In this case, an atomic concentration (stress) gradient in the wiring due to EM occurs, and the resulting atomic diffusion acts to cancel the atomic diffusion due to EM. It is said that the wiring life is prolonged. However, an EM damage dominant parameter for a wiring having a protective film in consideration of the above has not been specified yet, and a prediction method for a disconnection location and a disconnection life using the parameter has not been developed. On the other hand, damage to the protective film due to an increase in stress due to EM causes a short-circuit failure between wirings. However, no evaluation method has been developed for this reliability (short-circuit portion, life).
[0003]
[Problems to be solved by the invention]
The present invention relates to a factor that affects EM damage when a protective film is present, that is, an atomic concentration (stress) distribution in a wiring caused by EM, a thermal stress in a wiring caused by a protective film, and a diffusion coefficient (activity). The purpose is to formulate the governing parameters of EM damage theoretically in consideration of the change in the energies, and to develop a method for determining necessary physical constants and thermal stress using the parameters. It is another object of the present invention to provide a method for predicting a disconnection and a method for predicting breakage of a protective film using the governing parameters.
[0004]
[Means for Solving the Problems]
In order to achieve the above object, the present invention provides a device or a method for predicting damage to a metal wiring having a protective film on its surface, wherein a numerical analysis method such as finite element analysis is used to determine the current density and temperature distribution of the metal wiring, The current density and temperature distribution, the atomic concentration distribution, the physical flux constant of the wiring material, and the internal stress of the wiring, the atomic flux divergence of each element that divides the evaluation target into a void formation contribution and a hillock formation contribution. It is characterized by predicting the location of a short-circuit or disconnection fault and the time to failure by repeating this operation by separately calculating and determining the change in atomic concentration of each element based on the obtained atomic flux divergence of each element. And
These target metal wirings can be applied to either polycrystalline wirings or bamboo wirings.
Accordingly, it is possible to accurately predict short-circuit and disconnection damage of a metal wiring having a protective film on the surface.
The physical property constant of the wiring material and the thermal stress of the wiring are determined by using a linear wiring, setting the substrate of the wiring at a constant temperature, inputting a constant current for a certain period of time, and calculating the volume of the formed void. Contains.
As described above, the physical property constant of the wiring material and the thermal stress of the wiring can be easily obtained.
A recording medium storing a program for causing a computer to execute the metal wiring damage prediction method is also an aspect of the present invention.
[0005]
BEST MODE FOR CARRYING OUT THE INVENTION
Embodiments of the present invention will be described in detail with reference to the drawings.
The present invention provides protection for two types of interconnects having different internal structures, namely, a polycrystalline interconnect having a plurality of crystal grains in the interconnect width direction and a bamboo interconnect having only a single crystal grain in the interconnect width direction. EM damage dominant parameters in consideration of various effects caused by the presence of a film are specified, and a method of predicting disconnection and damage of a protective film in a polycrystalline wiring and a bamboo wiring is developed using each of them.
[0006]
<EM damage control parameter AFD_gen ^RFormulation>
First, the EM damage control parameter AFD_gen ^RThe formulation of will be described.
The influence factor of the protective film, that is, the EM atomic flux J in consideration of the back diffusion caused by the atomic concentration gradient and the thermal stress in the wiring is given by the following equation.
(Equation 1)

Here, N: atomic concentration (density), D₀: Frequency term, k: Boltzmann constant, T: absolute temperature, Q: activation energy, Z^*: Effective charge number, e: unit charge, ρ: electric resistivity, j: current density vector, σ_T: Tensile thermal stress, Ω: atomic volume, N₀: Atomic concentration (density) in no stress state, N_T: Σ_TIs the atomic concentration (density) when κ acts, κ: bulk modulus.
[0007]
(Polycrystalline wiring)
The diffusion path of atoms in a polycrystalline interconnect may be within the crystal grain boundary and within the crystal grain, but the mobility of atoms within the crystal grain is negligibly small compared to that at the grain boundary, so only the divergence at the crystal grain boundary is considered. think of. In order to consider the crystal grain boundaries in the polycrystalline wiring, a model in which the crystal grain boundary structure is averaged is introduced. FIG. 1 is an example of a model of a polycrystalline wiring structure. As shown in FIG. 1, a unit-thickness square including only one triple point formed of three crystal grain boundaries having a length of (√3) / 6 times the average crystal grain size d. Assume the elements. The area of the element is (√3) d²/ 4. It is assumed that grain boundaries II and III are symmetric with respect to grain boundary I and their narrow angles are close to 120 °, but there is a slight deviation 2Δψ.
[0008]
Let the x and y components of the current density vector j be j_x, J_y, The atomic concentration is N, the x-direction component and the y-direction component of the atomic concentration gradient are ∂N / ∂x, ∂N / ∂y, respectively, and the angle between the grain boundary I and the x-axis is θ. The current density, temperature, atomic concentration, and atomic concentration gradient at the ends of I, II, and III are represented as shown in FIG. By substituting these into equation (1), the atomic flux along the crystal grain boundaries at the respective grain boundary edges can be obtained. The direction going out of the element is defined as positive here. By multiplying the atomic flux at the edges of the grain boundaries I, II and III by the width δ and the unit thickness of the grain boundaries, the number of movements of atoms per unit time at each grain boundary edge is obtained. Add. After ignoring the small term and simplifying the equation using the current conservation law, the volume of the element (要素 3) d²Divide by / 4. In this manner, when the angle between the grain boundary I and the x-axis is θ, the number of reduced atoms per unit time and unit volume in the grain boundary diffusion, that is, the divergence AFD of the atomic flux_{gb θ} ^RIs given by
(Equation 2)

Where C_gb ^R= NδD₀/ K. AFD_{gb θ} ^RRepresents the number of atoms reduced per unit time and unit volume due to the EM affected by the atomic concentration gradient, that is, the divergence of the atomic flux. The first and second terms in <> on the right side of equation (2) are terms relating to the divergence of the atomic flux at the grain boundary triple point, and the other terms are the terms of the atomic flux at the grain boundary itself. This is a term related to divergence. Also, AFD_{gb θ} ^RA positive value contributes to the formation of voids (voids), while a negative value contributes to the formation of hillocks (projections on the surface). When an actual wiring is considered, θ takes an arbitrary value. Therefore, it is necessary to calculate the divergence of the flux in consideration of the entire range of θ (0 <θ <2π). Here, assuming only the void formation, the AFD when θ changes from 0 to 2π_{gb θ} ^RThe expected value of only the positive value of is calculated. Here, AFD_{gb θ} ^RIs considered to be 0 because it does not contribute to void formation. AFD_{gb θ} ^RAnd AFD_{gb θ} ^ROf the atomic flux related to void formation at the crystal grain boundary of the polycrystalline wiring by dividing the sum of the absolute values of_gen ^RIs derived as in the following equation.
(Equation 3)

[0009]
(Bamboo wiring)
On the other hand, in the bamboo wiring, the divergence of the atomic flux at the crystal grain boundaries can be ignored.Therefore, considering only the divergence within the crystal grains, the number of reduced atoms per unit time and unit volume, that is, Flux divergence AFD_lat ^RIs represented by the following equation.
(Equation 4)

Here, the current conservation law is used, and simplification is performed ignoring minute terms. AFD temporarily contributing to void formation_lat ^RIf only the positive value of_gen ^RIs given by the following equation.
(Equation 5)

[0010]
<Method of deriving physical property values>
(Polycrystalline wiring)
Parameter AFD governing EM damage in polycrystalline wiring considering the effect of protective film_gen ^RThe constants relating to the physical properties of the thin film contained in the following are the average crystal grain size d, the constant Δψ relating to the relative angle between the crystal grain boundaries, and the activation energy Q in crystal grain boundary diffusion._gb, Effective charge number Z^*, C_gb ^R= NδD₀/ K. The separation of these constants will be described below for the central portion of the straight wiring which can be regarded as having a constant current density and temperature and easy to handle.
First, N ≒ N at the center of the linear wiring_TI can go. Here C_gb ^R= NδD₀/ K is originally a variable related to N, but considering the stress value that the wiring can take, N of N₀Is considered to be at most several%, so that approximately N = N_T≒ N₀If you go, C_gb ^R= N₀δD₀/ K, which can be regarded as a physical property value of the thin film.
The concentration gradient ∂N / ∂x is inversely proportional to the wiring length (see Blech, IA, J. Appl. Phys., 47-4 (1976), 1203) and depends on the wiring shape. Value. Here, the concentration distribution is regarded as linear at the central portion of the wiring, and κ · ∂N / ∂x is derived as a constant depending on the wiring shape. For Δψ, a value derived from a wiring having a surface exposed simultaneously manufactured under the same conditions is used. In order to obtain the current density, current density gradient, temperature, and distribution of the temperature gradient of the wiring required for deriving the physical property values, numerical analysis such as finite element analysis is performed on the wiring used in the experiment.
[0011]
First, the average crystal grain size d is measured by a focused ion beam (FIB) device or the like. Activation energy Q_gb ^*[= Q_gb−σ_TΩ] is determined as follows. The same input current density j_lFor three different substrate temperatures T_s1, T_s2And T_s3An EM acceleration experiment in which current is supplied for a certain period of time is performed. Here, the substrate temperature T_s1The temperature at the center of the wiring at_l, T_s2When is T₂And T_s3When is T₃And the condition at the center of the wiring under each experimental condition is Condition 1: j₁And T₁, Condition 2: j₁And T₂, Condition 3: j₁And T₃And In addition, j₁Input current density j different from₄By adjusting the substrate temperature with respect to₂Condition 4: equal to₄And T₂However, an EM acceleration experiment in which electricity is supplied for a certain time is performed. After energization, the void volume near the center of the wiring is measured for each. AFD at the center of a linear wiring ignoring terms relating to temperature gradient, current density gradient and gradient of concentration gradient_gen ^RIs displayed for the conditions 1, 2, and 3. Each is multiplied by the area of the void volume measurement area, the thickness of the wiring, and the net energizing time and atomic volume excluding the incubation period from the energizing time. Since these and the actually measured void volume are equal, for example, the void measurement value at the center of the wiring under the condition 1 is V₁Then, in this case, the following equation is established.
(Equation 6)

here
(Equation 7)

(Equation 8)

A is the area of the void volume measurement area, Q_gb ^*= Q_gb−σ_TΩ, ρ₁Is the temperature T₁Of the wiring at t, t₁Is the net energization time excluding the incubation period from the energization time under condition 1, and thick is the thickness of the wiring. An equation similar to equation (6) holds for conditions 2 and 3. By taking the natural logarithm on both sides in equation (6) and subtracting the equation in condition 3 from the equation in condition 1, Q_gb ^*And the effective charge number Z^*And the relational expression (9) of the atomic concentration gradient κ∂N / ∂x is obtained.
(Equation 9)

here
(Equation 10)

ρ₃Is the temperature T₃Of the wiring at t, t₃Is the net energization time obtained by subtracting the incubation period from the energization time under condition 3. Here, T₁And T₃Σ between_TWas considered as minute and ignored. An expression similar to Expression (9) holds for Conditions 1 and 2 and Conditions 2 and 3. On the other hand, when the temperature at the wire center is the same and the current density is j₁, J₄By subtracting an equation similar to equation (6) for conditions 2 and 4 different from^*And κ · ∂N / ∂x are obtained.
[Equation 11]

Where ρ₂Is the temperature T₂Of wiring at V, V₂And V₄Is the measured void volume under conditions 2 and 4, respectively, t₂And t₄Is the net energization time obtained by subtracting the incubation period from the energization time under conditions 2 and 4. Substituting equation (11) into equation (9), Q in conditions 1 and 3_gb ^*(Q_gb13 ^*) And κ · ∂N / ∂x are obtained as in equation (12). Note that when the threshold current density of EM damage can be experimentally obtained, the condition 4 is not required and the same Z^*And κ · ∂N / ∂x, which can be used instead of equation (11).
(Equation 12)

Since an expression similar to Expression (12) holds between Conditions 1 and 2 and Conditions 2 and 3, Q_gb ^*And κ∂N / ∂x are obtained. Where Q_gb ^*Can take the same value in each equation, so the Q between the three equations_gb ^*Deviation (Q_gb13 ^*−Q_gb23 ^*)²+ (Q_gb23 ^*−Q_gb12 ^*)²+ (Q_gb12 ^*−Q_gb13 ^*)²By determining an appropriate κ · ∂N / ∂x such that Q becomes minimum, Q_gb ^*And κ · ∂N / ∂x are obtained. Then Z^*Is obtained by substituting the obtained κ · ∂N / ∂x into equation (11). Further, the constant C_gb ^RFor the conditions 1, 2, and 3, the measured value of the void volume on the left side in the equation (6) and the AFD on the right side_gen ^RIs determined so that the void volumes obtained by the numerical calculation using are the same. Thus, for each of conditions 1, 2 and 3,_gb ^RIs obtained, the average of these values is C_gb ^RValue.
When estimating the thermal stress inside the wiring, the Q obtained by the above method is used._gb ^*[= Q_gb−σ_TΩ] and Q obtained using the wiring without the protective film_gbFrom the value of_TCan be estimated.
As described above, all the physical property values can be obtained from a simple experiment using a linear wiring.
[0012]
(Bamboo wiring)
EM damage dominance parameter AFD in bamboo wiring considering the effect of protective film_gen ^RConstant relating to the physical properties of the thin film contained in_lat, Z^*, D₀It is. A method for deriving these constants for the cathode portion of the straight-line bamboo wiring, which is the void forming area, will be described.
First, in equation (4), N ≒ N_TI can go. Here, the concentration distribution is assumed to be linear at the wiring cathode portion, and κ∂N / ∂x is derived as a constant depending on the wiring shape.
Activation energy Q_lat ^*[= Q_lat−σ_TΩ] is determined as follows. The same input current density j_lThree different substrate temperatures T_s1, T_s2And T_s3An EM acceleration experiment in which current is supplied for a certain period of time is performed. Here, the substrate temperature T_s1The temperature of the wiring cathode part at the time of₁, T_s2When is T₂and_Ts3When is T₃And the conditions in the wiring cathode section under each experimental condition are as follows: Condition 1: j_lAnd T₁, Condition 2: j₁And T₂, Condition 3: j_lAnd T₃And After energization, the void volume of the wiring cathode portion is measured for each. ∂²N / ∂x²AFD at the cathode portion of the linear wiring assuming = 0_gen ^RAre displayed for the conditions 1, 2 and 3, and are multiplied by the net energizing time and the atomic volume of the void volume measurement area, the thickness of the wiring, and the energizing time excluding the incubation period, respectively. By equating these values with the actually measured void volume, for example, the measured void volume value of the wiring cathode portion under condition 1 is V₁Then, the following equation is established.
(Equation 13)

Where A is the area of the void volume measurement area, ρ₁Is the temperature T₁Of the wiring at t, t₁Is the net energization time under condition 1, and thick is the thickness of the wiring. An equation similar to equation (13) holds for conditions 2 and 3. By taking the natural logarithm on both sides in Expression (13) and subtracting the expression in Condition 3 from the expression in Condition 1, Q_lat ^*Effective charge number Z^*And the relational expression of the atomic concentration gradient κ · ∂N / ∂x is obtained. By the way, when the current density is small, the atomic flux due to the EM and the back diffusion due to the concentration gradient are exactly balanced, and the apparent atomic flux becomes zero, so that no EM damage occurs. The bundles overcome the back-diffusion due to the concentration gradient, causing EM damage. The threshold current density at this time is j_thAnd When Equation (6) becomes zero at the cathode, Z^*And κ∂N / ∂x holds.
[Equation 14]

Where ρ₄Is j_thThe resistivity at the temperature of the cathode under action, j_thIs determined experimentally. Equation (14) is_lat ^*And the effective charge number Z^*And 濃度 · に N / ∂x and Q_lat ^*Is obtained as in the following equation.
(Equation 15)

Here, Q determined by conditions 1 and 3_lat ^*To Q_lat13 ^*I put it. Also, t₃Is the net energization time under condition 3. Since an expression similar to Expression (15) holds between Conditions 1 and 2 and Conditions 2 and 3, Q_lat ^*And κ · ∂N / ∂x are obtained. Here, κ∂N / ∂x should have the same value in each equation, and therefore, the deviation of κ∂N / ∂x between the three equations (κ∂N / ∂x₂₃−κ∂N / ∂x_Thirteen)²+ (Κ · ∂N / ∂x₂₃−κ∂N / ∂x₁₂)²+ (Κ · ∂N / ∂x₁₂One κ · ∂N / ∂x_Thirteen)²So that the minimum_lat ^*By defining_lat ^*And κ · ∂N / ∂x are obtained. Here, κ · ∂N / ∂x determined by conditions 1 and 3 is replaced by κ · ∂N / ∂x_ThirteenΚ∂N / ∂x determined by conditions 2 and 3 is replaced by κ∂N / ∂x₂₃Κ∂N / ∂x determined by conditions 1 and 2 is replaced by κ∂N / ∂x₁₂I put it.
Then Z^*Is obtained by substituting the obtained κ · ∂N / ∂x into the equation (14). Further, the constant D₀Is N_T= N₀And the void volume measurement value on the left side of the equation (13) and the AFD on the right side_gen ^RIs determined so that the void volumes obtained by the numerical calculation using are the same.
When estimating the thermal stress inside the wiring, the Q obtained by the above method is used._lat ^*[= Q_lat−σ_TΩ] and Q obtained using the wiring without the protective film_latΣ from value_TCan be estimated.
As described above, all physical property constants can be obtained from simple experiments using linear wiring.
[0013]
<Simulation of disconnection process>
In the simulation according to the present invention, the wiring to be predicted is divided into elements and the thickness of each element is determined by AFD._gen ^RIn this case, the disconnection process is simulated by reducing the value based on. Here, in the case of polycrystalline wiring, in consideration of the growth form of voids that grow in a slit shape along the crystal grain boundaries, the wiring is arranged based on the average crystal grain size and average slit width of the wiring obtained from experiments and observations. The element thickness is reduced only by the slit-shaped element. On the other hand, in the case of bamboo wiring, it is not necessary to install a slit-shaped element. FIG. 2 shows an example of element division of the polycrystalline wiring 100.
The simulation process of the disconnection process will be described with reference to the flowchart shown in FIG. 3 taking the case of polycrystalline wiring as an example. First, the current density distribution and the temperature distribution of the wiring are obtained by numerical analysis such as finite element analysis (S204). AFD of each element_gen ^RIs calculated using these distributions and the physical property values of the thin film previously determined from the acceleration test (S206). AFD of each element_gen ^RThe initial value of the atomic concentration used for the calculation in (1) is given at the beginning of this routine (S202). At this time, AFD contributing to void formation_{gb θ} ^RAFD focusing on the positive value of_gen ^RValue and AFD contributing to hillock formation_{gb θ} ^RAFD focusing only on negative values of_gen ^RCalculate for both values. The amount of change in the number of metal atoms of each element in one step of the simulation is based on the volume of the element and the time of one step in each AFD._gen ^RTo obtain a contribution to void formation and a contribution to hillock formation (S208). Further, using this, the density initial value N for each element is calculated.₀(S210). Next, by adding these, the density N of each element is obtained (S214), and the density distribution of the entire wiring is obtained. Here, the concentration value is a critical value N at which voids are formed._minThe critical value N below which the hillocks are formed_maxIs determined (S216), and if it is within this range (Yes in S216), the AFD is again performed using the obtained atomic concentration as the incubation period for EM damage._gen ^RCalculate the value and repeat the subsequent calculations. On the other hand, as the simulation proceeds, the concentration becomes N_minTo N_maxWhen the value exceeds the range (No in S216), the thickness of the slit-shaped element is reduced in accordance with the amount of decrease in the atoms calculated from the amount of change in the atomic concentration contributing to void formation as the progress period of the EM damage (S220). ). At this time, the thickness of the slit-shaped element is reduced by using the density change amount of the slit-shaped element and the elements adjacent to the slit-shaped element. However, in the case of bamboo wiring, this processing is not necessary.
[0014]
Where N_min, N_maxAnd the κ value are determined in advance as follows from the results of the accelerated test. First, the value of κ 配線 N / ∂x at the center of the wiring and the latent period under a certain condition are obtained by an acceleration test using the linear wiring. Next, only the portion related to the incubation period in the previous simulation is performed under the same conditions, and the value of κ is determined so that κ · ∂N / ∂x becomes equal after the incubation time obtained in the acceleration test. Further, the maximum value of the concentration N at that time is expressed as N_maxValue, minimum value is N_minValue. If the κ value is known in advance by numerical calculation, experimental measurement, or the like, a simulation for the incubation period obtained in the test is performed using this κ value, and the maximum value of the concentration N at that time is calculated as N_maxValue, minimum value is N_minValue. Conversely N_max, N_minIs known, a simulation of the incubation period is performed, and N after the incubation period is calculated._maxOr N_minCan be determined such that is consistent with a known value.
In the element whose thickness has decreased, it is considered that a void having a depth corresponding to the amount of decrease in thickness has been formed (S220). In consideration of the thickness of each element (S226), numerical analysis such as finite element analysis of current and temperature is performed again.
The above calculation is repeated, and a state in which an element having a temperature exceeding the melting point or an element having a thickness equal to or less than the threshold is connected over the wiring width is defined as a disconnection (S222), and the simulation is terminated (S224).
[0015]
<Simulation of short-circuit fault>
In the simulation of the short-circuit fault, only the incubation period is calculated in the process of the flowchart of FIG. 3 described above. That is, in step S216, the density N becomes N_maxThe calculation is terminated assuming that the protection film has been destroyed when the value exceeds.
N_maxThe value can be given in the same way as in the previous section, but if the protective film in the accelerated test is different from that of the prediction target or if the prediction accuracy is to be increased, the κ value and the protective film Calculate the fracture strength or fracture toughness, and calculate N from the stress value and κ value at that time._maxIs determined and given. Although the destruction of the protective film occurs before the short-circuit failure and is strictly different, the time of the destruction of the protective film is approximately the first approximation.
[0016]
The configuration of the present invention can be realized by reading out and executing the program from a storage medium storing a program for predicting EM damage of metal wiring having the protective film of the present invention. The recording medium includes a floppy disk, a CD-ROM, a magnetic tape, a ROM cassette, and the like.
Examples of the system used for predicting the EM damage of the metal wiring having the protective film of the present invention include not only a stand-alone computer system but also a client-server system including a plurality of systems. May be used.
[0017]
【Example】
Hereinafter, specific examples of the above-described derivation of physical property values and prediction of disconnection will be described.
<Derivation of physical properties included in dominant parameters in polycrystalline wiring>
FIG. 4 shows the dimensions and shape of the Al thin-film wiring 100 for which the physical property constants have been actually determined based on the above concept. As shown in FIG. 4, a sample was prepared in which an Al thin film wiring was formed on a Si substrate having an oxide film formed on the surface, and the surface was coated with a polyimide resin film. This wiring was subjected to an EM acceleration test according to the above-described method for deriving the physical properties. Two types of wirings were created, SampleL: width (W) 8.6 μm and length (L) 82 μm, and SampleS: width (W) 8.7 μm and length (L) 32 μm.
The EM acceleration test was performed using an apparatus 300 having a configuration as shown in FIG. In order to derive the activation energy, the temperature of the substrate 360 is held at one of three temperatures of 458, 473 and 485 K by using the hot stage 350. The current density in the wiring section 100 is about 5.5 MA / cm²In addition to inputting and outputting a constant current from the constant current source 310 for a certain period of time, the substrate temperature is set to 415K so that the temperature at the center of the wiring is the same as the temperature at the center of the wiring when the substrate temperature is 473K. Holding, the current density in the wiring portion is 8.5 MA / cm in SampleL.², 9.0MA / cm for SampleS²A constant current was input and output from the probes 372 and 374 for a certain period of time. For each condition, 25 test pieces were used. After all the currents had passed, the protective film was removed, and the central region of the wiring was observed with a field emission electron microscope (FE-SEM) 400. The total area of the voids was measured by image analysis of the obtained SEM image, and the void volume at the wiring center was estimated by multiplying the void area by the film thickness.
[0018]
EM damage dominance parameter AFD considering the effect of protective film_gen ^RIs applied to a wiring having a protective film on the surface, and AFD is applied to each of SampleL and SampleS._gen ^RDerivation of physical property values included in. Obtained physical property value Q_gb ^*, Z^*, C_gb ^RTable 1 summarizes the values of κ∂N / ∂x.
[Table 1]
Table 1 AFD_gen ^RPhysical properties included in

As can be seen from Table 1, regardless of the difference in shape between SampleL and SampleS, Q_gb ^*, Z^*, C_gb ^RWere almost the same. In other words, it can be said that each of them functions as a property value independent of the shape. It is said that the gradient of the stress (concentration) in the wiring is inversely proportional to the wiring length. Is about 1.2 times larger. Q_gb ^*And Z^*The value obtained for is between the values reported so far for the Al thin film and can be said to be valid.
As described above, AFD is applied to the wiring having the protective film on the surface._gen ^RWas applied, the effect of the protective film was accurately grasped, and a physical property value independent of the wiring shape and a concentration gradient depending on the shape could be accurately derived.
[0019]
On the other hand, AFD_gen ^RIs applied to a wiring having the same shape as a wiring having a protective film and having no protective film on the surface, the obtained activation energy Q_gb ^*(= Q_gb) Was 0.547 ev for a wiring having the same shape as SampleL (hereinafter, referred to as SampleLl), and 0.549 ev for a wiring having the same shape as SampleS (hereinafter, referred to as SampleSl). Comparing the activation energy of the wiring with the protective film and the activation energy of the wiring without the protective film, both the SampleL and SampleS of the wiring with the protective film are smaller. If the influence of the interface diffusion is large, the activation energy should be large. However, since the activation energy is small in the wiring having the protective film, the EM damage due to the interface diffusion accompanying the protective film coating is reduced. The effect is considered to be small.
Also, Q^* _gb= Q_gb−σ_TWhen thermal stress was estimated for each of SampleL and SampleS from Ω, the thermal stress obtained from each sample was 29 MPa for SampleL and 415 MPa for SampleS.
[0020]
(Disconnection prediction)
It is assumed that the Al wiring having a linear shape shown in FIG. As the physical property values of the test pieces, the values in Table 1 obtained by an acceleration test using the prepared test pieces were used. Here, κ used a generally known value.
In the disconnection prediction, an input current density of 5.5 MA / cm is used for the wiring shape having the value of SampleL shown in FIG.², Substrate temperature: 473K. FIG. 6 illustrates the result of the disconnection prediction. As shown in FIG. 6, damage is caused on the cathode side. In the prediction by this simulation, a disconnection at 35415 sec near the wiring cathode portion was predicted.
As described above, it was possible to predict the life of the wiring having the protective film and the disconnection location.
[0021]
【The invention's effect】
According to the present invention described above, highly accurate prediction of disconnection (disconnection location, life) for a wiring having a protective film, and prediction of damage (short-circuit location, life) of a protective film covering the wiring, in accordance with the actual wiring structure Becomes possible. Also, by using a method of determining physical constants included in EM damage dominating parameters for a wiring having a protective film, evaluation of thermal stress inside the wiring and diffusion coefficient (activation energy) due to interaction between the protective film and the wiring. Can be evaluated, and it becomes possible to select a protective film and a wiring material in designing an integrated circuit and to give an index of strength evaluation.
[Brief description of the drawings]
FIG. 1 is a diagram showing an example of a model of a polycrystalline wiring structure.
FIG. 2 is a diagram showing an example of element division of a polycrystalline wiring.
FIG. 3 is a flowchart showing a simulation process of a disconnection process in the case of a polycrystalline wiring.
FIG. 4 is a diagram showing a dimensional shape obtained from physical property constants.
FIG. 5 is a diagram showing an apparatus configuration for performing an EM acceleration test.
FIG. 6 is a diagram illustrating a result of disconnection prediction.

Claims (6)

In a metal wiring damage prediction device having a protective film on the surface,
Means for determining the current density and temperature distribution of the metal wiring by a numerical analysis method such as finite element analysis,
Based on the obtained current density and temperature distribution, the atomic concentration distribution, and the physical property constant of the wiring material and the wiring internal stress, the atomic flux divergence of each element that divides the object to be evaluated is defined as a void formation contribution and a hillock formation contribution. Means to divide into
Means for calculating the change in atomic concentration of each element based on the obtained atomic flux divergence of each element, and predicting the location of the short-circuit or disconnection failure and the time until the failure by repeating the operation of each means. An apparatus for predicting damage to metal wiring. 2. The damage predicting device according to claim 1, wherein the metal wiring is a polycrystalline wiring or a bamboo wiring. In a method for predicting damage of a metal wiring having a protective film on a surface,
Obtaining a current density and a temperature distribution of the metal wiring by a numerical analysis method such as a finite element analysis;
According to the obtained current density and temperature distribution, the atomic concentration distribution, and the physical constants of the wiring material and the wiring internal stress, the atomic flux divergence of each element that divides the evaluation target into a void formation contribution and a hillock formation contribution. Steps to ask separately,
Determining a change in the atomic concentration of each element based on the obtained atomic flux divergence of each element, and by repeating each step, the location of the short-circuit or disconnection failure and the time until the failure are predicted. To predict damage to metal wiring. 4. The damage prediction method according to claim 3, wherein the metal wiring is a polycrystalline wiring or a bamboo wiring. 5. The method for predicting damage to metal wiring according to claim 3, wherein the physical constant of the wiring material and the internal stress of the wiring are set to a constant temperature and a constant current for a fixed time using a straight wiring. A method of predicting damage to a metal wiring. A recording medium storing a program for causing a computer to execute the metal wiring damage prediction method according to claim 3.

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