Abstract:As one of the important factors affecting the reliability of Complementary metal oxide semiconductor integrated circuits, the time-dependent dielectric breakdown of gate oxide has always been the focus of domestic and foreign scientists. The previous researches have shown that when the electrons pass through the silicon dioxide lattice and collide with it, part of their kinetic energy will be dissipated, then among of which can create some sort of defects that behave as electron traps. However, when the electron traps density reaches a critical value in the gate oxide, the destructive thermal effects open a low-resistance ohmic path between the electrodes, thus, triggering the breakdown conducting mechanism. In this paper, in order to employing a statistical theoretical analysis method, the behavior related with time-dependent dielectric breakdown of gate oxide was investigated in detail. The following useful results are obtained, (i) According to the microscopic mechanism of the time-dependent dielectric breakdown of gate oxide under electrical stresses associated with the randomness and irreversibility of the electron traps generation in gate oxide, a theoretical analysis method has been proposed, which is based on the equation of the electron traps generation rate combined with the Langevin equation in random theory and the Fokker-Planck equation in the non-equilibrium statistical theory. (ii) In light of the dynamic equilibrium model, the generation rate equations of electron traps under constant current stress and constant voltage stress as well as the probability density distribution functions of electron traps density were determined. By integrating these distribution function, the cumulative failure rate was further obtained. (iii) Taking the specific metal oxide semiconductor capacitors as examples, the relationship between the most probable lifetime of the gate oxide and the constant current stress, constant voltage stress, as well as its thickness have been revealed. And the concept of "breakdown limit" was defined by analogy with the concept of "fatigue limit" in the phenomenon of solid fracture. (iv) The dependence of the cumulative failure rate on the current stress, voltage stress, and time has been presented. A characteristic value of the time was introduced to indicate the time when the cumulative failure rate reaches 0.63. At this time, most devices have breakdown failure. Besides, the test parameters are not affected by previous failures, and they are close to that of the cumulative failure rate of 1. The results show that the probability density distribution function of electron traps density satisfies lognormal distribution, and the obtained distribution of failures fits well with the experimental statistical data. Keywords:gate oxide/ time-dependent dielectric breakdown/ nonequilibrium statistical theory/ cumulative failure
其中$\dot N\left( t \right)$为t时刻电子陷阱的生成速率; $k\left( N \right)$为电子陷阱的平均生成速率, 是电子陷阱生成速率的确定部分, 由平均结构和施加的电应力共同决定; $\beta \left( N \right)$为涨落放大函数, $f\left( t \right)$为涨落函数, 二者的乘积$\beta \left( N \right)f\left( t \right)$是电子陷阱生成速率的随机部分, 由不均匀性涨落决定. 假设电子陷阱的生成速率与达到的退化程度无关, 只与当时及稍早施加的电应力和栅氧化层的微观结构有关[11], 故可将电子陷阱的生成过程看作一个马尔可夫过程, 涨落函数$f\left( t \right)$服从如下高斯分布:
$ F = \int_{{N_{{\rm{bd}}}}}^\infty {P\left( {{N_0},{N_{\rm{j}}},t} \right)}{\rm{d}}{N_{\rm{j}}}. $
结合(15)和(12)式, 并代入数据计算可得某一恒流应力下的累积失效率. 图5和图6分别是样品1和样品2在不同恒流应力下, 累积失效率F随时间t的变化曲线. 图 5 样品1中两种不同电流密度的累积失效率 (a) J = 0.1 A/ cm2; (b) $J = 1\;{\rm{A}}/{\rm{c}}{{\rm{m}}^{2}}$ Figure5. Breakdown cumulative distributions for two different electric current densities in sample 1: (a) J = $ 0.1~\rm A/cm^{2} $; (b) $J = 1 \;{\rm{A}}/{\rm{c}}{{\rm{m}}^{ 2}}$
图 6 样品2中三种不同电流密度的累积失效率 (注: 实心五角星代表实验值, 线段代表理论值.) Figure6. Breakdown cumulative distributions for three electric current densities in sample 2. (Note: solid five-pointed star represents experimental value and line segment represents theoretical value.)
由(20)式可知, 不考虑临界电子陷阱密度${N_{{\rm{bd}}}}$随栅氧化层厚度l的变化, 最概然寿命${t_{{\rm{bd}}u}}$随样品厚度l的增大而增大, 随电场强度E0的增大而减小. 根据样品2的数据进行分析, 不同厚度的栅氧化层的最概然寿命${t_{{\rm{bd}}u}}$如图7所示；不同恒定电场E0下, 栅氧化层的最概然寿命${t_{{\rm{bd}}u}}$如图8所示. 从图8中可以看出, 栅氧化层的寿命${t_{{\rm{bd}}u}}$随电场强度E0的增大而减小, 其“击穿极限”大约为$ 5\; {\rm{MV}}/{\rm{c}}{{\rm{m}}} $, 且与实验结果[22]趋势一致. 图 7 不同厚度的栅氧化层的最概然寿命 Figure7. The most probable lifetime of gate oxide with different thickness.
图 8 不同电场强度对应的栅氧化层的最概然寿命 Figure8. The most probable lifetime of gate oxide under different electric field.
根据(19)式得到相同电场下, 不同时刻栅氧化层中电子陷阱密度的概率密度曲线如图9所示; 以及同一时刻, 不同电场下的概率密度曲线如图10所示. 由图9和图10可以看出, 随电场强度E0和时间t的增大, 电子陷阱密度的概率密度的最概然值逐渐增大, 且电子陷阱密度的概率密度分布趋于均匀. 图 9${E_0} = 13.75 \;{\rm{MV}} \cdot {\rm{c}}{{\rm{m}}^{ - 1}}$时, 不同时刻的P-N图 Figure9. The probability density varies with electron trap density at different time under ${E_0} = 13.75\; {\rm{MV}} \cdot {\rm{c}}{{\rm{m}}^{ - 1}}$.
图 10$t = 30 \;{\rm{s}}$时, 不同电场强度下的P-N图 Figure10. The probability density varies with electron trap density at $t = {\rm{30\; s}}$ under the different electric field.
根据(15)式可得某一恒定电场强度E0下的累积失效率F随时间t变化的图像, 并与实验结果[22]进行了对比, 如图11所示. 当K2 = 100 MV/cm时, 理论结果与实验结果拟合较好, 但实验得到${K_2}$的值一般在200 MV/cm左右[24]. 可以看出, 当电场强度E0为13.7 MV/cm, 14.00 MV/cm 和14.25 MV/cm时,$\tau $分别为71.8 s, 46.8 s和30.7 s, 即随电场强度E0增大栅氧化层的寿命变短; 随时间t和电场强度E0的增大, 累积失效率F逐渐增大, 且理论结果与实验结果相符. 图 11 三种不同电压下的累积失效率 Figure11. Breakdown cumulative distributions for three different electric field.
图 1 不同厚度栅氧化层的最概然寿命 (a) 样品1[21], 
图 2 不同电流密度对应的栅氧化层的最概然寿命 (a) 样品1; (b) 样品2
图 3 不同时刻的P-N图 (a) 样品1, 
图 4 不同电流密度下的P-N图(
图 5 样品1中两种不同电流密度的累积失效率 (a) J = 0.1 A/ cm2; (b) 
图 6 样品2中三种不同电流密度的累积失效率 (注: 实心五角星代表实验值, 线段代表理论值.)
图 7 不同厚度的栅氧化层的最概然寿命
图 8 不同电场强度对应的栅氧化层的最概然寿命
图 9 
图 10 
图 11 三种不同电压下的累积失效率