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Sensors and Actuators A 133 (2007) 173179

Principles of space-charge based bi-stable MEMS: The junction-MEMS

Jean-Michel Sallese , Didier Bouvet

Ecole Polytechnique Federale de Lausanne (EPFL), Institut de Microelectronique et Microsysteme, Laboratoire dElectronique Generale,

CH 1015 Lausanne, Switzerland

Received 11 November 2005; received in revised form 15 March 2006; accepted 22 March 2006

Available online 11 May 2006

Abstract

We propose a theoretical investigation of a new principle of bi-stable MEMS based on the remnant electrostatic force generated by the space-

charge density that builds up when two semiconductors, having different doping densities, are brought into contact. Following a physical descriptionof the device, a complete analytical formulation of some key parameters such as the pull-in and pull-out voltages could be obtained. The model

reveals that both closed and open configurations of the switch MEMS can be stable states at zero applied voltage. The impact of the different

technological parameters is also discussed in details, supporting that the so-called junction-MEMS might be an interesting alternative to address

bi-stability in MEMS, still remaining fully compatible with the standard microelectronics technologies.

2006 Elsevier B.V. All rights reserved.

Keywords: MEMS; RF; Space-charge; Bi-stability; Switch; Junction

1. Introduction

Switch and RF MEMS are still receiving a lot of interest due

to their foreseen unique applications in RF signal processing aswell as in low power consumption solutions [110]. Among

the different structures of interest are the so-called bi-stable

MEMS, since they offer the possibility to memorize a state

without a continuous power supply. This ability is very attractive

for some specific applications such as in reconfigurable circuit

architecture [10,11]. Different approaches were proposed so far

to achievea bi-stable operation.Withoutentering into the details,

we can mention that most of the solutions presented so far are

based on a mechanical bi-stability relying on the concept of

buckling beams [1217]. Such deformations can be achieved

whether by electrostatic, thermal or, more unexpected, optical

induced displacements [1820]. Bi-stability was also obtainedthrough magnetic-based solutions [2123]. Unlike precedent

devices,thesedo not rely on a mechanical nature of bi-stability,

but they require the integration of a coil and relatively high cur-

rent pulses are needed to actuate the device.

Recently, an electrostatic based bi-stable solution was sug-

gested by the authors who proposed to use the hysteresis of

Corresponding author. Tel.: + 41 21 693 46 02; fax: +41 21 693 36 40.

E-mail address: [email protected] (J.-M. Sallese).

the polarization in ferroelectrics materials to generate two sta-

ble states [24]: a remnant charge on the plates of a classical

MEMS capacitor can generate an electrostatic force without any

applied potential. Following the same motivation, we propose anew principle to generate electrostatic bi-stability in MEMS

that is compatible with standard CMOS microelectronic tech-

nology, where typical process steps are illustrated in [2,3,25,26].

The idea consists of creating a standard MEMS capacitor where

at least one electrode is a doped semiconductor. For the struc-

ture of interest, we will treat the particular situation where these

are, respectively, P and N type semiconductors. In closed mode,

the built-in space-charge density, subsequent to the formation

of the depletion region in the PN junction, will act as the rem-

nant electrostatic force. However, other combinations can also

be used. In particular, one electrode can be a metal or any con-

ducting layer that gives rise to space-charge when brought incontact with the semiconductor surface. The aim of this work

is to investigate this junction-MEMS from a theoretical point of

view.

2. General analysis

The structure of interest is depicted on Fig. 1. It consists of

a standard capacitor MEMS with N and P type semiconduc-

tors electrodes that can also be covered by an insulating layer,

0924-4247/$ see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.sna.2006.03.024
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174 J.-M. Sallese, D. Bouvet / Sensors and Actuators A 133 (2007) 173179

Fig. 1. Cross section of the junction-MEMS structure.

depending on the specific application that is pursued. A springof stiffness k linked to the moving electrode tends to keep the

system open. An attractive electrostatic force can be created

by applying a potential V on the electrodes. This in turn will

modulate the gap between the plates. The corresponding energy

band diagram of such structure is shown on Fig. 2.

The total force acting on the moving electrode is a combina-

tion of the spring and electrostatic forces [24]:

FTot =S2

20 k(dv0 dv) (1)

where the first and the second terms in the RHS of (1) rep-

resent, respectively, the electrostatic and mechanical forces. In

this expression, S is the electrode surface, the charge density

per unit area (assumed positive in the model), dv0 and dv the gaps

between the electrodes when these are, respectively, neutral and

charged. Before merging electrical and mechanical descriptions,

we propose to evaluate the electrode charge density as a function

of the applied potential and electrodes spacing.

Fig. 2. Sketch of the energy band structure of the junction-MEMS depicted on

Fig. 1.

2.1. Derivation of the electrodes charges densities

From [27], an explicit relation exists between the semicon-

ductor charge density (per unit surface) and the potential drop

at its surface, usually called the surface potential, n and pin the present study (Fig. 2). Assuming that the intrinsic carrier

density is negligible with respect to the doping concentration,

we can express the surface charge densities in the P and N type

semiconductors [27]:

QP = sign(p)

2scUTqNA

ep/UT+

p

UT 1

+

n2i

N2A

ep/UT

p

UT 1

1/2(2a)

QN = sign(n)

2scUTqND

n2

iN2D

e

n/UT+

n

UT 1+

en/UT n

UT 11/2

(2b)

where ND and NA are the doping concentration of the N and

P type semiconductors, UT the thermal voltage (25.8 mV at

300 K), ni the intrinsic semiconductor density, q the electron

charge and sc the semiconductor dielectric constant.

These relations are valid for depletion,inversion and accumu-

lation modes (see [27] for a detailed discussion). Since depletion

will be of major concern in this work, we propose to illustrate our

structure by the corresponding energy band diagram as shown

on Fig. 2 (note that the sign must be changed for the potential

counterpart). Insulating layers (we suppose that both have thesame dielectric constant i) of thicknesses diN and diP as well

as space distribution of fixed charge densities (per unit volume)

n(x) and p(x) [27] are also included.

Defining as the charge density per unit area of the N type

electrode, we can write from the charge neutrality principle:

= QN + n = (QP + p) (3)

where n,p =

diN,iPn,p(x) dx represent the total fixed charge

density per unit surface.

In addition, when a potential V is applied, we must ensure

that the following relations are satisfied:

V= Vn Vp =EFp EFn

q(4)

V = p n + (n p)

+

diN

n(x)

id2x

diP

p(x)

id2x

+dv

0+QN

diN

iQP

diP

i(5)

where n,p(x)/id2x holds for the integral form of the Pois-

son equation evaluated over the corresponding dielectric layers

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J.-M. Sallese, D. Bouvet / Sensors and Actuators A 133 (2007) 173 179 175

(assuming integration constants to zero), CiP and CiN the insula-

tors capacitanceson topof theP andN semiconductor electrodes,

EFp and EFn the Fermi levels of the P and N electrodes and

p and n the energy differences between the Fermi levels and

the intrinsic level [27] (see Fig. 2). Furthermore, noting that

(p n) is the built-in potential Vbi of a pn junction [27]

given by Vbi = p n = UT ln(NAND/n2

i

), we can define an

effective built-in potential Veffbi as:

Veffbi = Vbi

diN

n(x)

id2x

diP

p(x)

id2x +

p

CiP

n

CiN

(6)

This effective built-in potential depends on local distribution

of the fixed chargetraps inside the insulating layers. It is interest-

ing to note that Veffbi can also be negative, which highlights that

the location of the fixed charges densities can probably affect

the overall device properties. Defining the total insulator capac-itance Ci by 1/Ci = 1/CiN + 1/CiP, relation (5) becomes:

V+ Veffbi = p n +

dv

0+

1

Ci

(7)

Having these relations in hand, we are now able to address

the stability problem of the junction-MEMS.

2.2. Stability criteria of the global structure

Instabilityin classical switch MEMSwas already addressed

by different authors [28,29]. However, due to the presence

of semiconductor electrodes, the situation needs to be care-fully analyzed. Basically, equilibrium requires that FTot =0 and

d(FTot)/d(dv) > 0, i.e. static equilibrium and energy minimiza-

tion conditions must be satisfied. Merging these conditions with

relation (1) leads to the following inequality [23]:d()

d(dv)+

2(dv0 dv)

> 0 (8)

We still need to determine how the semiconductor charge

density varies with the gap between the electrodes while the

potential V is maintained constant. Differentiating relation (7)

with respect to dv withdV= 0, and assuming thatthe fixed charge

density is independent of the potential [27], we have:

0 =

dn

dQN

dp

dQP+

1

Ci+

dv

0

d+

0

d(dv) (9)

Formally, the quantitiesdQN,P/dn,p canbeidentifiedasthe

semiconductor intrinsic capacitances noted CN and CP, respec-

tively. Using (9) into (8), we obtain a critical distance between

the semiconductor electrodes that can be considered as the

threshold of instability (i.e. the pull-in effect):

dpull-in =2

3d0

0

3

1

Cpull-in

N

+1

Cpull-in

P

+1

Ci

(10)

Cpull-inN and C

pull-inP are the intrinsic capacitances evaluated at the

threshold of instability, which are not already determined at this

stage of the derivation.

The second condition must ensure that the electrostatic and

mechanical forces compensate each other (FTot = 0 in relation

(1)). This gives the critical charge density pull-in in terms of

technological parameters and intrinsic capacitances:

2pull-in =2

30

k

S

d0 + 0

1

Cpull-inN

+1

Cpull-inP

+1

Ci

(11)

Eq. (11) represents an implicit relation in terms of pull-insince the intrinsic capacitances depend on the surface potentials

at threshold, i.e. pull-inp and

pull-inn , and thus are also dependent

on pull-in through relations (2a) and (2b). At this point of the

derivation, no assumption is made about the magnitude of the

surface potentials: this analysis is thus valid in all modes of

operation.

Once the critical charge density pull-in is known, the elec-

trode separation at the onset of instability is readily obtained

through relation (10). This in turn leads to the determination of

the pull-in voltage (Vpull-in) which is of major importance for the

design of switch MEMS:

Vpull-in = pull-inp

pull-inn + pull-in

dpull-in

0+

1

Ci

Veffbi

(12)

To go one step further towards an analytical formulation and

gain some insights from the model, we propose to distinguish

two major operating modes: accumulation and strong inversion

on one side, and depletion on the other side.

In the first situation, i.e. when both electrodes are whether

in strong inversion (si) or in accumulation (acc), the intrinsic

capacitances are found to be quite high since the dependence

of charges versus surface potential is mainly exponential (see

relations (2a) and (2b)). In this case, relations (10) and (11)

can be simplified, leading to an asymptotic critical distance and

critical charge density:

dpull-inacc,si=

2

3d0

0

3Ci(13)

2pull-inacc,si=

2

30

k

Sd0 +0

Ci (14)

The similarity between these asymptotic relations with those

derived for metallic electrodes (but now with insulating layers

that can also have fixed charges) is indeed consistent since the

very high carrier concentration at the semiconductor surfaces act

as metal plates. However, since n and p are not negligible,

the pull-in voltage will differ from the classical expression

Vpull-in = pull-indpull-in/0.

Conversely, once the electrodes are in contact, we canexpress

the so-called pull-out potential (Vpull-out) below which the elec-

trodes will separate. Criterion to achieve a non-contact mode is

that the mechanical force must overcome the electrostatic force

when dv = 0. From relations (1), we obtain the pull-out charge

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density pull-out:

pull-out =

20

k

Sdv0 (15)

Then, inserting (15) into (7) and using (3) in combination

with (2a) and (2b) gives the potential drop in the semiconductor

electrodes, i.e. pull-outn and pull-outp . The pull-out voltage is thenreadily obtained:

Vpull-out = pull-outp

pull-outn +

pull-out

Ci Veffbi (16)

Note that if we are dealing with strong inversion, the sur-

face potentials pull-inp,n or/and

pull-outp,n reach asymptotic values

[27] given by p,nsi2UT ln(NA,D/ni). Then, pull-in andpull-out

potentials can also be fairly well approximated by an analytical

expression.

2.3. Asymptotic expressions with electrodes in depletionmode

Before starting the approximate derivation, let us recall that

depletion requires that the surface potential must be lower than

twice the potential difference between the intrinsic and Fermi

levels [27], i.e. |p,n| < 2UT ln(NA,D/ni).

Under normal operation, the electrodes are found to be in

depletion when instability occurs. This observation allows us to

derive simple analytical expressions for the different quantities

of interest. Assuming surfaces in depletion at the threshold of

instability (pull-in), relations (2a) and (2b) can be simplified as

follows:

QP

2scqNAp (17a)

QN

2scqND(n) (17b)

which in turn leads to analytical expressions for the intrinsic

capacitances:

1

CN=

dn

dQN

QN

scqND=

n

scqND(18a)

1

CP=

dp

dQP

QP

scqNA=

+ p

scqNA(18b)

Note that in order to satisfy the depletion assumption, we

must ensure that the critical charge density is higher than the

fixed charge density, i.e. n >0 and + p >0 in all the

situations, i.e. for pull-in and pull-out conditions. Defining an

equivalent doping concentration Neq and an equivalent insulat-

ing layer capacitance Ci eq (Ci eq can be negative by definition)

as:

1

Neq=

1

ND+

1

NA(19a)

1

Ci eq=

1

Ci+

p

scqNA

n

scqND(19b)

Solving relation (11) with (18a) and (18b) gives the critical

charge density as a function of technological parameters only:

pull-in =20

3scqNeq

k

S

+ 2

03scqNeq

k

S2

+2

3 02

k

Sdv0

0+

1

Ci eq

(20)

which in turn leads to the critical distance dpull-in as a function

of the critical charge density:

dpull-in =2

3d0

0

3

pull-in

scqNeq+

1

Ci equi

(21)

At this point, it is interesting to note that the pull-in charge

density increases when the equivalent doping concentration Neqdecreases. In some cases, the critical distance can also be neg-

ative, meaning that there is no more instability in the structure.

This can be achieved whether by decreasing the equivalent insu-

lating capacitance or equivalent doping density.

The threshold potential of instability is obtained from rela-

tions (17a), (17b) and (12):

Vpull-in =(pull-in n)

2

2scqND+

(pull-in + p)2

2scqNA

+ pull-in

1

Ci+

dpull-in

0

Veffbi (22)

Conversely, from relations (15) and (16), we obtain an

approximate expression of the pull-out potential Vpull-out

:

Vpull-out =(pull-out n)

2

2scqND+

(pull-out + p)2

2scqNA

+pull-out

Ci Veffbi (23)

Relations (22) and (23) have a uniquepropertyas compared to

the classical formulas sinceVpull-in and Vpull-out can take negative

values.

A negative Vpull-in means that even with no applied volt-

age, due to the built-in potential (that depends on the doping

densities), electrodes can move spontaneously in contact. This

situation has to be avoided for the application of interest. Then,

we must ensure Vpull-in > 0. A worst case is obtained when both

electrodes are highly doped, leading to (note that we still must

make sure that the depletion approximation holds):

0

k

S

2

3

1

Ci+

d0

0

3/2> Veffbi (24)

when Veffbi is negative, this relation is always satisfied.

Thefact that Vpull-out canbe negative is themain conclusion of

this work. A negative pull-out voltage means that, even with no

applied potential (V= 0), once the electrodes are in contact, the

open configuration can be recovered only by setting the potential

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to a negative value: the electrodes will remain in contact at zero

applied voltage, which is a signature of bi-stability.

Two asymptotic situations can then be discussed:

If the effective doping is very high, while there is a finite

insulator capacitance, we get:

20

k

Sdv0 < Veffbi Ci (25)

Conversely, when both the insulating layers and fixed charge

densities are negligible (in regard to pull-out), we can write:

1

qNeq

0

sc

k

Sdv0 < V

effbi (26)

Finally, when the integrated fixed charge densities n and nare negligible with respect to pull-out, the condition Vpull-out < 0

is given by:

1 < scqNeq 1

Ci eq+ 1

C2i eq+2 V

eff

biscqNeq

20 kS

dv01/2

(27)

Relations (25)(27) can be used to properly choose the tech-

nological parameters to design the junction-MEMS. However,

we might need a more intuitive parameter that depicts how

stable is the closed mode of the device under V= 0. For practical

applications, it might be interesting to evaluate the electro-

static to mechanical force ratio when the device is in closed

mode (V= 0). We propose to call this ratio the stability factor

(SF):

SF =2contact

20dv0

S

k

(28)

where contact is the electrode charge density when V=0 and

dv = 0. In the most general situation, this can be solved self con-

sistently by combining relations (7) with (3), (2a) and (2b). If

we deal with the depletion approximation, we can propose an

approximate solution mainly valid when fixed charge densities

n and p are negligible with respect to contact (this has to be

verified a posteriori):

contact = scqNeq

1Ci eq

+

1C2i eq

+ 2Veffbi

scqNeq

(29)Further interpretation of the pull-in and pull-out voltage

dependences with device parameters is not straightforward. In

the next section, we propose to perform some illustrative simu-

lations.

3. Numerical application

In this section, we assume that the semiconductor of interest

is silicon and the insulating layer is silicon dioxide.

3.1. Negligible fixed charge density

Based on existing switch MEMS published in the liter-

ature, we propose to investigate two different structures by

choosing a k/S factor of 5109 and 2 1010 N m3 (respec-

tively, k= 5 0 N m1, S=100 100m2 and k=200Nm1,

S=100 100m2), while varying the doping concentration

keeping NA =ND. Note that the latter k/S is very high and is

mainly used to estimate the degree of bi-stability of the device.

The oxide thickness on each electrodes was 10 nm and the elec-

trode separation dv0 was 1m.

Fig. 3 shows both the pull-out and the stability factor as a

function of the semiconductor doping concentration, for the two

cases of interest. Not shown on the graphic are the pull-in volt-

ages that are, respectively, close to 12.5 and 27 V. These values

are found to be nearly independent of the doping concentrations

and were not added on the plots.

Concerning the pull-out voltages, we find as expected that

the higher the k/Sfactor, the higher Vpull-out. For the case corre-

sponding to k/S= 5 109, the pull-out voltage becomes negativeas soon as thedoping concentration exceeds 11016 cm3.This

means that at zero applied voltage, the electrodes cannot sepa-

rate since the remnant electrostatic force starts to dominate the

mechanical counterpart. Further increase in the doping concen-

tration leads to a saturation ofVpull-out to a value close to Veffbi ,i.e.

Vbi in this case. However, the electrostatic to spring force ratio

greatly increases, thus giving a much more stable situation when

doping is increased. These observations are quite interesting and

suggest that it is always possible to separate the electrodes with

potentials as low as 1V (for silicon) provided fixed charge den-

sities are small. This in turn open the way to realize a simple

MEMS device with two stable states at zero applied potential.The same situation occurs for the higher k/S ratio. Now the

threshold for achieving a negative pull-out potential is pushed

Fig. 3. Simulation of the pull-out voltage and electrostatic to spring force ratio

as a function of the semiconductor doping concentration (we assume NA =ND),

for different values ofk/S(open circles: k/S= 5 109

; dots: k/S= 21010

).

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towards higher doping concentration, i.e. around 5 1016 cm3.

Same conclusions hold for the stability factor. This result is

interesting since it confirms that the junction-MEMS is indeed a

reliable bi-stable system even when the spring stiffness reaches

quite high values. In addition,it comes outthat thepull-in voltage

is almost independent on the doping concentration and oxide

thicknesses when n and n are negligible. Its value mainlydepends on the electrode gap separation and on the value of k/S.

Then, pull-in and pull-out voltages can be sized independently,

which is very interesting to relax some design constraints.

3.2. Impact of fixed charge density on critical parameters

Fixed chargedensity in silicon dioxide hasbeen the subject of

extensive research (see [27] for a general discussion). It comes

out thatthesechargesare almost positive (weakly correlatedwith

to the silicon doping) and located at the Si/SiO2 interface. How-

ever, the definition of the effective built-in potential (relation

(6)) clearly states that the spatial distribution of fixed charges

will directly affect the device characteristics.Then, we propose to evaluate different situations correspond-

ing to p =0and n =1012 cm2, whether assuming that traps

are located at the Si/SiO2 interface or at the bare surface of the

SiO2 layer. The semiconductors doping densities were also var-

ied (still keeping NA =ND), and the k/Sratio was set to 5109.

Results are plotted on Fig. 4. Full circles and triangles repre-

sent, respectively, n =1012 cm2 and n =+10

12 cm2. Dot-

ted lines stand for traps located at the Si/SiO2 interface, whereas

full lines hold for traps located at the bare surface of the insu-

lating layer. Obviously, both the sign of the fixed charges and

their location deeply impact Vpull-out. When the silicon doping

Fig. 4. Simulation of the pull-out voltage. Full circles: n =1012 cm2, trian-

gles: n =+1012 cm2 assuming p = 0 in both cases.Dotted lines standfor traps

located at the Si/SiO2 interface, whereas full lines correspond to traps located

on top of the SiO2 layer.

concentration exceeds 3 1017 cm3, the lower pull-out volt-

age is obtained for a negative charge density n =1012 cm2

(majority carriers for the N type silicon) and when the charge

are on the bare SiO2 surface. On the contrary, the lower Vpull-outis obtained for n = 10

12 cm2, still with traps located at the

surface. Intermediate values of Vpull-out correspond to charges

trapped at the Si/SiO2 interface. Not shown here is the pull-involtage which value, close to 12 V, was found to be almost insen-

sitive to the trap density when NA,D exceeds 31017 cm3.

Such observations can serve as general guidelines for the design

of the junction-MEMS. Relatively high doping densities should

be used to realize a bi-stable system. In addition, in case of

charge injection, it is better to use the device in depletion mode

where injection of majority carriers (with respect to the semi-

conductor type) on top of the dielectric layer can emphasize the

bi-stable nature of the junction-MEMS. Finally, the effect of the

fixed charges can be minimized if very thin insulating layers are

used.

4. Conclusion

In this work, we investigated a unique feature of a new con-

cept of a space-charge based bi-stable MEMS. This device relies

on the built-in charge density that builds up when a doped semi-

conductor on one side and an adequate conducting layer on the

other side are brought in contact. In particular, when both mem-

branes consist of P and N type doped semiconductor, general

relations could be derived providing guidelines for a correct

choice of physical parameters to achieve bi-stability. Interesting

aspects of such junction based bi-stable MEMS rely on its low

power capabilities and process compatibility with the state of

the art CMOS technology.

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Biographies

Jean-MichelSallese receivedthe diploma of engineer fromthe Institut National

des Sciences Appliquees (France) in 1988 and the PhD in physics from the

University of Nice-Sophia Antipolis (France) in 1991, where he worked on deep

levels characterization in IIIV semiconductors. He joined the Swiss FederalInstitute of Technology in Lausanne (EPFL) in 1991 where he was involved

in IIIV laser diodes characterization and inter-diffusion in quantum wells and

quantum wires structures. He currently gives lectures in semiconductor devices

and his research activities concern compact modelling of bulk and multigate

MOSFETs as well as modelling of MEMS and ferroelectric based devices.

Didier Bouvet obtained hisPhD thesisfrom theSwissFederalInstitute of Tech-

nology (EPFL) in 1996, where he worked in the field of thin films nitridation

for nonvolatile memories. After being a process engineer at ATMEL Company

from 1996 to 1998, he joined the Electronic Laboratories of the EPFL as a

research associate, where he developed new slurries for CMP applications in

close collaboration with microelectronics industry. More recently, his research

activities also concern CMOS process optimization dedicated to nano technol-

ogy as well as development of new MEMS structures. He is author or co-author

of 20 scientific publications and 5 patents.

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