本文先建立用偏移循環加工法及推導出加工奈米流道梯型凹槽到預定深度與寬度所需的預估偏移加工的加工道次及底部凸起的公式，並建立加工梯型凹槽到預定深度及寬度的方法。由於切削的深度會直接影響切削寬度，故本文先決定好凹槽深度，而之後再處理預定之凹槽寬度。由於梯型凹槽寬度與偏移量之間又有關聯性，故本文將探針，看作為一球體，而利用其相同凹槽深度切削道次之橫切面，可將其形狀作為相交之圓形。因此用此原理本文推導出偏移量、凹槽深度及凹槽寬度之關係式。利用此關係式代入所預定之凹槽深度及寬度，能得出探針所需之總偏移量。將總偏移量來分做不同的切削道次，再將其切削道次代入凹槽底部凸起量計算公式，得出凹槽底部凸起量的上凸值。本文並提出上凸值應收斂在所設定之目標收斂值0.54nm內。假如估算之中間偏移量之切削道次所計算出的底部上凸值超出目標收斂值，則再多加一切削道次，直到估算之中間偏移量的切削道次其梯型凹槽底部上凸值可收斂在目標收斂值。
　　而一般在做AFM奈米流道梯形凹槽切削時，通常都會在快切到接近目標深度時，再改變一次下壓力來使得最後一層之深度接近目標深度。然而因實際AFM機台在改變下壓力時，改變下壓力的時間約為7分鐘左右，因此此方法會多了一次需要改變下壓力的時間，故還不算是較有效率的方法。為了在實際應用上取得最少加工前置時間，本文提出加工奈米流道梯形凹槽到預定深度及寬度的較少加工道次及改變下壓力次數的加工方法及其目標函數和限制條件。本文一開始先定探針加工最大下壓力的安全係數，在安全係數下的下壓力開始模擬加工梯型凹槽之深度，然後逐步調整下壓力，模擬加工梯型凹槽之加工深度，使其逐漸逼近梯型凹槽之預定深度。而確定接近梯型凹槽預定深度之後，則在每一切削層第一切削道次皆設定此下壓力值，第二切削道次則改變下壓力取得與第一切削道次之相同切削深度，最後應用比下壓能理論模式進一步估算出可達到預定奈米流道梯形凹槽深度，進而能達到最少改變下壓力的次數。而探針加工到預定之梯型凹槽寬度，則應用上面所述的加工到預定寬度方法。可加工到預定的梯型凹槽之寬度。進而估算出所需加工道次次數，使其有較少的加工道次次數。
　　在這裡為了防止探針在切削多次時造成探針因疲勞而產生斷裂，因此我們設了安全係數，得出了在安全係數下之最大下壓力，再利用此下壓力逐步慢慢調整下壓力大小，使其下壓力能夠加工逼近到所預定之深度。然而本文第一切削層之預定切削深度可以用比下壓能之方法來直接得出所預定之切削深度，故可略過逐步逼近之步驟。而本文之第二切削層與第三切削層切削之後的加工預定深度，則無法直接利用比下壓能公式來預估此深度，故這時須利用最佳化之逐步逼近法來反覆模擬調整下壓力，進行第二切削層切削或第三切削層切削使其在設定之下壓力加工奈米流道之梯型凹槽深度能剛好逼近所預定深度。加工到預定深度確定了之後再來利用上述之方法調整偏移量，來達到加工梯型凹槽到預定寬度。針對梯型凹槽之不同預定深度與寬度，也能比照上述方法模擬出滿足較少的加工道次次數及改變下壓力次數的下壓力。

The paper uses offset cycle cutting method and derives equations of required cutting path and upward height at the bottom in the estimated offset cutting for cutting of trapezium groove to the expected depth and width on nanochannel, and establishes a method for cutting of trapezium groove to the expected depth and width. Since cutting depth would directly affect cutting width, the paper firstly determines the depth of groove, and then deals with the expected width of groove. Since the width of trapezium groove is associated with offset amount, the paper treats the probe as a sphere, and uses the cross-section of cutting path with the same groove depth, and its shape can serve as an intersecting circle.
Therefore, using this principle, the paper derives a relational equation among offset amount, groove depth and groove width, and then substitutes the relational equation in the expected groove depth and width to obtain the required total offset amount of the probe. The paper divides total offset amount into different cutting paths, and then substitutes the cutting paths in the calculation equation of upward height at the bottom of the groove to achieve the upward height value at the bottom of the groove. The paper also suggests that the upward height value should converge at below the preset objective convergence value 0.54mm. If the upward height value at the bottom calculated from the cutting path of the estimated middle offset amount exceeds the objective convergence value, one more cutting path can be added again and again until the upward height value at the bottom of the trapezium groove at the cutting path of the estimated middle offset amount can converge at the objective convergence value.
In general cutting of trapezium groove on nanochannel by atomic force microscopy (AFM), when cutting is conducted and it is getting close to the objective depth, down force is usually changed once to make the depth on the last layer close to the objective depth. Nevertheless, when down force is actually changed on AFM, the time for change of down force is around 7 minutes.
Therefore, this method, requiring more time to change down force once more, is not considered a more efficient method. In order to achieve the least lead time of cutting during actual application, the paper proposes a cutting method requiring less cutting paths and less number of change of down force for cutting trapezium groove to the expected depth and width on nanochannel, as well as the objective function and constraints. First of all, the paper sets a safety factor of maximum down force for cutting by probe. With down force under the safety factor, simulation of cutting of a groove to a depth begins. Then down force is step by step adjusted, and cutting of trapezium groove to a cutting depth is simulated, making it gradually approximate to the objective depth of trapezium groove.
After it is ascertained that the objective depth of trapezium groove is almost reached, this down force is set for the first cutting path on each cutting layer. In the second cutting path, down force is changed to achieve the same cutting depth as that in the first cutting path. Finally, specific down force energy (SDFE) theoretical model is applied to further estimate that the expected depth of trapezium groove on nanochannel can be achieved, and the least times for change of down force can be achieved.
When the probe has cut the trapezium groove to the expected width, the abovementioned method for cutting to the expected width can be applied so as to cut the trapezium groove to the expected width. Furthermore, the required number of times of cutting path can be estimated, so that the number of times of cutting path can be less.
In order to prevent the probe from being broken due to fatigue after cutting for multiple times, we have set a safety factor, and achieve the maximum down force under the safety factor. This down force is used to step by step adjust the down force size slowly in order to make it available to approximate to the expected depth at the down force. Nonetheless, for the expected cutting depth on the first cutting layer, the paper can use SDFE method to directly achieve the expected cutting depth. Thus, step by step approximation steps can be skipped over. As to cutting of the expected depth after cutting on the second cutting layer and the third cutting layer, SDFE equation cannot be directly used to estimate such depth. Therefore, during this time optimal step by step approximation method has to be employed to repeatedly simulate adjustment of down force and conduct cutting on the second cutting layer and cutting on the third cutting layer, so as to exactly approximate to the expected depth during cutting of trapezium groove depth on nanochannel at a set down force.
After it is ascertained to have cut to the expected depth, the above method can be used again to adjust the offset amount, so as to cut the trapezium groove to the expected width. Focusing on different expected depths and widths of trapezium groove, the above method can also be used to simulate the down force that satisfies the needs of less number of times of cutting path and less number of change of down force.

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