• Keine Ergebnisse gefunden

Antenna design

Im Dokument Charge sensing (Seite 50-53)

The antenna rules as presented in Section 2.3 are similar to the antenna principle presented in this section. The principle from this section can be used in an additional, electrostatic influencing way. The antenna structures are targeted to “collect” charges during the application of a process step. Their collected charge is then transferred to a (small) test transistor which is measured afterwards for certain parameters.

Within this work, charge is also collected through such an “antenna” structure and is conducted to the charge sensor. Additionally, the charge sensor is very sensitive and can detect image charges already. This is not the case with the conventional

“antenna rule” verification structures from technology design from Section 2.3. As a consequence, the influence from charges near to the antenna is discussed in this section. Those charges that are collected by the antenna are conducted through the terminal and do not need further consideration.

Charge can be detected through the presence of electrical field. Obviously this would also allow to measure the electrical field itself if no charge is present. In order

h g



q antenna surrounding

Figure 4.6: One dimensional problem for the explanation of mirror charge in the antenna charge measurement setup.

4.3. ANTENNA DESIGN 51 to do so, the following simplified (one dimensional) setup is discussed, see Figure 4.6.

A conductor is connected to the charge sensor input and is placed at a distanceg apart from a chargeq. This conductor is called “antenna” in this work. The term

“antenna” stems from the fact that the conductor receives the electrical field of nearby charge, as laid out in the following. There is no connection to radio frequency (RF) signal reception, despite the name similarity.

The space between the charge and the antenna is filled with a dielectric with constantεr. The remaining space is filled with air or vacuum (εr ≈1) and at a distance h, the surrounding enclosure is again a conductor with connection to “ground”. The setup is unbound in two dimensions making for a one dimensional electrostatic problem, where the field is along this dimension. Within this setup, the integral form of the third Maxwell equation, Equation 1.2, allows to calculate the field strength from the charge amounts. With the areaAof the setup in the two homogeneous dimensions, the third Maxwell equation is evaluated “around” the chargeq:

A ∂D

∂x A qV

rε0Einsulator+Aε0EvacuumA q (4.4)

The antenna is connected to the charge sensor input, which resembles a virtual ground.

As both, the antenna and the surrounding conductor are at ground potential, the integral ofEmust zero out if going from surrounding to the antenna:

h Evacuum g Einsulator

Due to the presence of the electric field in the insulatorEinsulator, a mirror chargeq builds up on the antenna surface. This mirror charge must be conductively supplied by the charge sensor. The mirror chargeqis again calculated using Maxwell’s equation, whereEinsulatoris negative due to the inverted direction of the setup compared to the previous case ofq. The divergence is evaluated at the antenna surface (aroundq):

−Aεrε0EinsulatorA q (4.5)

Substitution into Equation 4.4 yields

−A q−Aε0

g h

q εrε0 A q

The amount of charge loss 1− q

q specifies how good−q approximatesq and is calculated as:

1 + q

q g

r+g (4.6)

In order to measureqwith low charge loss (i.e. −q≈q),εr andh should be large andg should be small. Onceεrh ≫ g is satisfied1, the error is negligible and the charge measurement resemblesqeven though no conductive connection between charge and charge sensor is established.

The simple one dimensional setup represents a real physical setup with limited extension into all directions, too. Except for the simple requirement forh andd, the requirement is extended into three dimensions: For all objects close to the charge, the direct distance between the charge and the object must be significantly larger than the direct distance to the conductor connected to the charge sensor. If this is still fulfilled, the charge measurement of the charge sensor output resembles the negative amount of charge under investigation:−q.

The Shockley-Ramo theorem [16, 17] was developed to ease the complex analysis of image charges in real physical setups to yield the current value from moving of such charges. A means to calculate the currentIin the case that the charge e moves (⃗v) relative to a conductor (e.g. the antenna) is, according to [17, equation (1)]:

Ie⃗Ev·⃗v (4.7)

The field⃗Evin this equation contains the information on the physical setup. It is the field at the position of e if the antenna is at unit potential and all other conductors are set to zero potential. The scalar product of⃗Ev and⃗v resembles the influence of the field from the charge onto the antenna. In this theorem, thecurrentin the antenna terminal resulting from the charge movement in vicinity of the antenna is expressed. If the charge reaches the antenna, the conduction of charge must be applied, Equation 1.3, if the charge does not reach the antenna but settles to a steady position (i.e. stops moving), the previous considerations about the image charge have to be applied. The Shockley-Ramo theorem affects the currentbeforethe charge reaches a steady position in relation to the antenna (i.e. a displacement current). In all subsequent analyses where current is derived, the observation of current according to the Shockley-Ramo theorem can be neglected. The time-frame from the point where the charge influences the antenna (i.e. theE⃗vterm from Equation 4.7 becomes significant) up to the time when the charge reaches a steady position in relation to the antenna is small compared to the sampling time of the measurement recording. This is applicable to all measurements conducted within this work. As a consequence, the current can be simply calculated as derivative of the charge amount on the antenna, according to the image charge analysis from this section.

With an antenna, “remote” charge sensing can be implemented, independent of the type of material that contains the charge. This can be an insulator (see Section 6.2 for results) or a conductor, such as a needle (see Section 5.6 for examples). In all cases, charge can be transferred to field once the setup is known sufficiently well, by the Maxwell equations. As discussed in Section 4.2, charge can be directly applied to the charge sensor through a conductor. This allows the usage as very sensitive

1Note the opposite sign ofqandq, which requires Equation 4.6 to go to zero, hence the inequality.


Im Dokument Charge sensing (Seite 50-53)