Capacitive MEMS-based sensors : thermo-mechanical stability and charge trapping

Micro-Electro Mechanical Systems (MEMS) are generally characterized as miniaturized systems with electrostatically driven moving parts. In many cases, the electrodes are capacitively coupled. This basic scheme allows for a plethora of specifications and functionality. This technology has presently matured and is widely employed in industry. \ud A voltage across the electrodes will attract the movable part. This relation between electric field and separation (or capacitance) can be conveniently employed to sense tiny displacements (less than 1 pm) or for example as a through sensor for electromagnetic power through a coplanar waveguide (CPW).\ud The design involves often dielectric layers, whether accidental (native oxide) or intentional. During fabrication and / or operation of the device, trapped charge can uncontrollably accumulate and decumulate in these layers, causing parasitic forces on the device. These parasitic forces can influence the device beyond acceptable limits. The research described in this thesis approaches this phenomenon in two ways: a) device level and b) fundamental level. \ud a) Device level: Complete MEMS structures. The thesis contains theory of capacitive MEMS, including amongst others pull-in voltage, electrostatically loaded clamped-clamped beams, and electro-mechanical resonance, as well as the origin, dynamics and influence of trapped charges, in conjunction with built-in voltage. A cryogenic experimental study has been done on the effect of charge trapping on two types of MEMS-based RF power sensors. New structures have been realized with far better thermo-mechanical immunity. These structures are the first to involve double beam springs, which are fabricated by wet KOH etching of silicon. It is demonstrated that even an ultrathin aluminum oxide (native, ~ 2 nm) can harbor significant charge trapping. The dynamics are found to slow down considerably at cryogenic temperatures. At last, a study is done on when charge trapping actually limit the performance of real MEMS devices: a gravity gradiometer and an RF power sensor. \ud b) Fundamental level. Existing measurements and imaging of local trapped charges by conducting atomic force microscopy (AFM) are reinterpreted by a new model. This multi mirror model calculates the electrostatic interaction (force, gradient and contact potential difference) between the sample surface containing the trapped charge and the tip of the AFM, represented by a conducting sphere. This model improves drastically over existing calculations. Some interesting theoretical approximations of quantities describing this interaction have been found

Images of Locally Finite Derivations of Polynomial Algebras in Two Variables

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and $div D=0$ is a Mathieu subspace of $k[x, y]$.Comment: Minor changes and improvements. Latex, 9 pages. To appear in J. Pure Appl. Algebr

The tame automorphism group in two variables over basic Artinian rings

In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption is really needed: we will give, for local Artinian rings with square-zero principal maximal ideal, a complete description of the tame automorphism subgroup. This will lead to an example of a non-tame automorphism, for any characteristic p>0.Comment: 10 page

Stable Tameness of Two-Dimensional Polynomial Automorphisms Over a Regular Ring

In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.Comment: 18 page

Two Results on Homogeneous Hessian Nilpotent Polynomials

Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be {\it Hessian Nilpotent}(HN) if its Hessian matrix \Hes P(z)=(\frac {\partial^2 P}{\partial z_i\partial z_j}) is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $\Delta^m P^{m+1}(z)=0$ for any $m>>0$.} In this paper, we first show that, the VC holds for any homogeneous HN polynomial $P(z)$ provided that the projective subvarieties ${\mathcal Z}_P$ and ${\mathcal Z}_{\sigma_2}$ of $\mathbb C P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $\sigma_2(z):=\sum_{i=1}^n z_i^2$, respectively, intersect only at regular points of ${\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z-\nabla P$ with $P(z)$ HN if $F$ has no non-zero fixed point $w\in \mathbb C^n$ with $\sum_{i=1}^n w_i^2=0$. Secondly, we show that the VC holds for a HN formal power series $P(z)$ if and only if, for any polynomial $f(z)$, $\Delta^m (f(z)P(z)^m)=0$ when $m>>0$.Comment: Latex, 7 page

Venereau-type polynomials as potential counterexamples

We study some properties of the Venereau polynomials b_m=y+x^m(xz+y(yu+z^2)), a sequence of proposed counterexamples to the Abhyankar-Sathaye embedding conjecture and the Dolgachev-Weisfeiler conjecture. It is well known that these are hyperplanes and residual coordinates, and for m at least 3, they are C[x]-coordinates. For m=1,2, it is only known that they are 1-stable C[x]-coordinates. We show that b_2 is in fact a C[x]-coordinate. We introduce the notion of Venereau-type polynomials, and show that these are all hyperplanes, and residual coordinates. We show that some of these Venereau-type polynomials are in fact C[x]-coordinates; the rest remain potential counterexamples to the embedding and other conjectures. For those that we show to be coordinates, we also show that any automorphism with one of them as a component is stably tame. The remainder are stably tame, 1-stable C[x]-coordinates.Comment: 15 pages; to appear in J. Pure and Applied Algebr