The linear coefficient for the thermal expansion of quartz used for a mask plate is 5. 5×10-7 oC-1. This implies that when the mask plate is heated or cooled, its length changes by an amount proportional to the original length, the change in temperature, and the linear coefficient for thermal expansion of quartz. Mathematically, we can describe this relationship with the following equation: where T is change in the temperature of the mask plate, l is the change in length of the mask plate caused by the thermal expansion.
is the linear coefficient for the thermal expansion of quartz, and l0 is the largest distance from the portion of the mask plate that remains aligned during expansion to any portion on the mask plate that becomes misaligned. Therefore, ? = 5. 5×10-7 oC-1. Alignment accuracy across a 6 inch (15 cm) silicon substrate must be maintained from one layer to the next within 0. 5 um. The mask must be properly aligned on the wafer. However, if there is a change in the temperature of the mask plate during alignment, the mask plate will expand or contract.
The amount that the mask plate expands or contracts will be determined by the above equation. All of the features on the mask plate will also expand or contract in accordance with the above equation. I have solved this question with the assumption that a 1:1 exposure system, such as a vacuum hard contact, soft contact or proximity system, was used. Therefore, a change in length of one of the features on the mask plate will create an identical change in the pattern that the mask creates on the silicon substrate.
In order to ensure that alignment accuracy across the silicon substrate is maintained from one layer to the next within 0.5 um, none of the features on the mask plate must expand or contract more than 0. 5 um. Therefore, the change in length of the mask plate caused by thermal expansion must be less than or equal to 0. 5 um. Let us take ? l to be the largest allowable change in length of the mask plate. This yields, ? l = 0. 5 um. Therefore, in the above equation for ? l, we need to ensure that the left side of the equation, which represents the largest allowable change in length of the mask plate, is always greater than or equal to the right side, which represents the actual change in length of the mask plate.
Mathematically, This equation can be rearranged to solve for ? T in the following manner. As stated previously, a change in length of the mask plate will create an identical change in the pattern that the mask creates on the silicon substrate. It is not stated in the question exactly how the mask is aligned to the silicon substrate. In other words, we are not told what portion of the mask remains fixed in position to the silicon substrate during thermal expansion, and what portion does not remain fixed.
Therefore, I have solved the remainder of the question under the assumption that one edge of the mask remains aligned with one edge of the silicon substrate after the thermal expansion. This implies that the features on the extreme opposite edge of the mask, which are initially separated from the fixed edge by a distance of 15 cm, experience the greatest alignment error due to thermal expansion. However, if I had assumed that the centre of the mask remained aligned with the centre of the silicon substrate during expansion, I would have received the same answer.
Both ends of the mask would simply have expanded in equal and opposite directions, producing a misalignment in one direction at one edge, and an equal misalignment in the other direction at the other edge. Therefore, the total alignment error would have been the same regardless of where the mask was aligned to the silicon substrate. Under the assumption that one edge of the mask remains aligned with one edge of the silicon substrate after the thermal expansion, l0 must equal 15 cm. Using the conversion factor 1 cm = 10-2 m and 1 um = 10-6 m, we can convert l0 to um in the following manner:
With significant figures applied Substituting our values for l0, ? l and ? into the above equation for ? T yields: with significant figures applied This implies that the maximum allowable change in mask plate temperature during alignment is 6. 060606061oC. Therefore, regardless of the initial temperature of the mask plate, the maximum allowable range of mask plate temperature is 6. 060606061oC around that initial temperature. 4-2). According to the question, we may assume that a negative resist molecule is a straight chain of CH2 units.
The molecular mass of one molecule of a compound can be determined by adding together the molar masses of each of the atoms in that compound. There is one atom of carbon and two atoms of hydrogen in one molecule of CH2. Therefore, the molecular mass of one molecule of CH2 can be determined by the following formula: where MMCH_2 is the molecular mass of one molecule of CH2, MMC is the molar mass of one atom of carbon, and MMH is the molar mass of one atom of hydrogen. The molar mass of one atom of carbon is 12 amu, while the molar mass of one atom of hydrogen is 1 amu.
Substituting these values into the above formula, the molecular mass of one molecule of CH2 can be calculated in the following manner: with significant figures applied The number of CH2 units in one negative resist molecule can be determined by dividing the total molecular mass of one negative resist molecule by the molecular mass of one molecule of CH2. Mathematically, this can be stated as: where NCH_2_units is the number of CH2 units in one negative resist molecule and MMresist is the total molecular mass of one negative resist molecule.
The negative resist molecule under consideration in this question has a molecular mass of 100000 amu. In order to satisfy the bonding requirements of all the hydrogen and carbon atoms in the molecule, the two carbon atoms at the extreme ends of the resist molecule should each be bonded to three hydrogen atoms, forming a CH3 unit at either end of the resist molecule. Therefore, in order to obtain the molecular weight of a chain containing only CH2 units, I have subtracted the atomic weight of these two hydrogen atoms.
This yields a value of MMresist = 100000 – 2 = 99998 amu. Substituting the values for MMresist and MMCH_2 into the above equation for NCH_2_units yields: with significant figures applied Note that the raw value for NCH_2 stated above has no physical meaning – there must be an integer number of CH2 units in one negative resist molecule. Therefore, I have completed the remainder of this question by making an assumption. I have rounded the number of CH2 units in one negative resist molecule to the nearest number of complete CH2 units.