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arXiv:1106.1480v1 [quant-ph] 8 Jun 2011

Offsets in Electrostatically Determined Distances: Implications

for Casimir Force Measurements

S. K. Lamoreaux∗and A. O. Sushkov†

Yale University, Department of Physics,

P.O. Box 208120, New Haven, CT 06520-8120

Abstract

The imperfect termination of static electric fields at semiconducting surfaces has been long known

in solid state and transistor physics. We show that the imperfect shielding leads to an offset in

the distance between two surfaces as determined by electrostatic force measurements. The effect

exists even in the case of good conductors (metals) albeit much reduced.

PACS numbers:

∗Electronic address: steve.lamoreaux@yale.edu

†Electronic address: alex.sushkov@yale.edu

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It has been long known that a static electric field applied external to a semiconductor will

not be perfectly internally shielded. We previously considered the case of Debye screening,

which predicts a penetration depth of 0.7 µm for Ge, and 24 µm for Si (intrinsic undoped

materials at 300 K) [1], which leads to distance offset when the force due to a voltage applied

between the plates is used for distance determination. The distance correction given by the

twice the Debye length divided by the (bare) dielectric constant. Our measurements with

Ge [2] did not support such a large distance correction, and indeed it has been long known

that surface states lead to a shielding of the internal field. The first attempts to build a field

effect transistor showed an internal field at least 100 times smaller than expected based on

bulk material considerations.[3] Bardeen provided a complete explanation of the importance

of surface states, and the basic premises are well established fundamental considerations in

transistor physics.[4] These effects largely explain why no distance correction was observed

in our Ge experiment. The analysis presented here, which included band bending and

surface state effects, shows that the simple Debye screening treatment of this problem is an

oversimplification.

We are reconsidering the problem because in our recent attempts at measuring the

Casimir force between high resistivity (> 10kΩcm) Si plates, we see a distance offset of

between 60 nm and 600 nm, depending on how the plates were cleaned. The sensitivity of

the effect to surface condition suggests that the surface states are likely the most important

part of the offset correction. In our study of the effect, it has become apparent that distance

offsets are probably affecting all Casimir experiments to date, and is an effect that needs to

be considered among the panoply of systematic effects that are only recently being acknowl-

edged. It is interesting to note that all of these systematic effects (roughness, vibration,

surface patch potentials, and now the offset effect considered here) all lead to an apparent

increase in the Casimir force compared to its true value at a given distance as determined

electrostatically.

The basic physical principle is shown and explained in Fig. 1. It is usually assumed

that the charge density in the depletion region is constant (nd) over its width d1.[5] This

approximation is very good in the case where there Fermi level is well below the conduction

band; raising the potential of the conduction band a small amount essentially empties it.

Shown in Fig. 2 are the related processes that are important for an electrostatic distance

calibration. We consider a case where the system is in equilibrium when a potential V0is

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applied between the plates, and then ask what happens when V0is changed by δV .

First, when δV = 0, we have (in units with ǫ0= 1, and the electron charge e = 1)

σ0d0+ V1= V0

(1)

by energy conservation and

σ0+ σ1− ndd1= 0(2)

by charge neutrality, where σ0 is the charge on the perfectly conducting plate, σ1 is the

surface state charge on the semiconducting plate, d0is the physical plate separation. Inte-

grating the electric field from the surface into the bulk, using Poisson’s equation, indicates

that

V1= αd2

1

(3)

where α ≈ nd/κ with κ the bare dielectric constant. Furthermore, we have

σ1= (EF− V1)ns

(4)

where nsis the surface density of states (e.g., electrons/cm2volt in appropriate units). We

thus find that

σ0= −(EF− V1)ns+ nd

?

V1/α(5)

or

σ0= −(EF− (V0− σ0d0)ns) + nd

?

(V0− σ0d0)/α (6)

which gives the relationship between σ0and V0which can be used to find the net effective

differential capacitance. It is easiest to change V0by δV and determine δV1, δd1, δσ0and

δσ1, with fixed nd, EF, ns, and d0. From charge neutrality, we have

δσ0− nsδV1− ndδd1= 0. (7)

Furthermore,

δd1=1

2

δV1

√αV1

(8)

and

d0δσ0+ δV1= δV. (9)

We thus arrive at

δσ0+ nsd0δσ0+ d0nd(4αV1)−1/2= nsδV + nd(4αV1)−1/2δV(10)

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which gives a differential capacitance (per unit area) of

C =δσ0

δV

=

ns+ nd(4αV1)−1/2

1/d0+ ns+ nd(4αV1)−1/2×1

d0. (11)

Evidently, this equation can be written in a form

C =

CACB

CA+ CB

(12)

where

CA=1

d0

= Cgap

(13)

and

CB= ns+ nd(4αV1)−1/2= Csurface+ Cbulk= Cmaximum=

1

doffset

(14)

which is the maximum capacitance that can be observed as d0 → 0. It should be noted

that the offset distance does not depend on d0; the capacitance between a spherical and

flat surface has the same offset as that for flat plates, in the limit that the proximity force

approximation applies.

Taking into account surface and bulk (space charge) effects shows that the net (differen-

tial) capacitance is lowered due to the parallel combination of Csurfaceand Cbulkin series

with Cgap. Thus, the plates are closer than the distance given by an electrostatic calibra-

tion. When both plates are made of the same semiconducting material, the distance offset

calculated here is simply multiplied by 2.

It is almost impossible to know the parameters required to calculate the surface and

bulk differential capacitances. For our recent measurements with Si, we have measured the

distance offset directly in the experiment (in situ) by performing an electrostatic distance

determination, then measuring how far the plates must be moved until they touch. A

more accurate determination of the offset was possible by firmly mounting the plates, with

one on a translation stage. The offset was determined by measuring the capacitance as a

function of distance until the plates touched, which could be readily electrically determined.

A maximum capacitance, at zero plate separation as determined by the electrical contact,

was directly measured, and found to be 62 ± 5 nm. After further cleaning of the plates, an

in situ measurement shows a 600 nm distance offset.

We further note that even in the case of good conductors, there is a space charge (bulk)

correction to the capacitance. In [6] (Sec. 3-2.4) it is shown that in the case of a parabolic

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conduction band, the effective distance offset is about 0.1 nm per plate. This means that

for distances of approximately 100 nm, in a force gradient type experiment between a flat

and smooth surface, noting that the electrostatic distance offset means the plates are closer

than expected, the correction is

δF

F

= −4δd

d

= −4 × (−0.2)/100 ≈ 1%(15)

which is at the level of the claimed accuracy of several experiments.

Because the offset is very sensitive to the physical surface properties, it would appear as

prudent to either directly measure the offset, or include it as an adjustable parameter in

comparing theory to experiment. We have found that it is simple to measure the maximum

capacitance that occurs just before the plates come into physical (electrical) contact. This

distance needs to be corrected by the surface roughness, and the measurements must be done

at sufficiently low frequency for equilibrium to be established, however, the measurements

are straightforward.

[1] S.K. Lamoreaux, arXiv:0801.1283 .

[2] W.-J. Kim, A.O. Sushkov, D.A.R. Dalvit, and S.K. Lamoreaux, Phys. Rev. Lett. 103, 060401

(2009).

[3] W. Shockley adn G.L. Pearson, Phys. Rev. 74, 232 (1948).

[4] J. Bardeen, Phys. Rev. 71, 717 (1947).

[5] A.J. Dekker, Solid State Physics (Prentice-Hall, Englewood Cliffs, N.J., 1957, ninth printing),

Sec. 14-4.

[6] Dietrich Meyerhofer, “Conduction through Insulating Layers,” in Field Effect Transistors (J.

Torkel Wallmark and Harwick Johnson, editors) (Prentice-Hall, Englewood Cliffs, N.J., 1966),

Sec. 3-2.4.

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