Sensing Mechanism of Nanowire Biosensors van der Lans.pdf · In chapter 3 only the in uence of the charge of the biomolecules is stud-ied. In chapters 4, and 5 the e ect of volume

Sensing Mechanism of NanowireBiosensors

Dorien van der Lans

March 19, 2009

Photo: Philips

Supervisors: dr. L.-F. Feinerdr. R. van Roij

Contents

Contents 3

1 Introduction to nanowire biosensing 5

2 Basics of biosensing with FETs 9

2.1 The �eld-e�ect transistor . . . . . . . . . . . . . . . . . . . . 10

2.2 Biomolecule properties . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Biomolecule attachment . . . . . . . . . . . . . . . . . 14

2.2.2 Surface Charge . . . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Sensor sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 The capacitance of a at metallic FET . . . . . . . . . 23

2.4.2 The capacitance of a cylindrical metallic FET . . . . . 23

2.4.3 The capacitances of a at and a cylindrical metallicFET compared . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 Screening in the at geometry . . . . . . . . . . . . . . 26

2.5.2 Screening in the cylindrical geometry . . . . . . . . . . 28

2.5.3 Comparing the double layers of the at and the cylin-drical geometries . . . . . . . . . . . . . . . . . . . . . 30

3 The Surface Charge Model 35

3.1 The Surface Charge Model for the at sensor . . . . . . . . . 37

3.1.1 Small potentials . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 Large potentials . . . . . . . . . . . . . . . . . . . . . 41

3.1.3 Comparing the small and the large potential regime . 42

3.2 The Surface Charge Model for the cylindrical sensor . . . . . 44

3.2.1 Small potentials . . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Large potentials . . . . . . . . . . . . . . . . . . . . . 47

3.2.3 Comparing the small and the large potential regime . 48

3.3 Comparing the e�ect of surface charge for the at and thecylindrical sensor . . . . . . . . . . . . . . . . . . . . . . . . . 48

3

4 CONTENTS

3.4 Applying a voltage to the sensor . . . . . . . . . . . . . . . . 51

4 The Excluded Volume Model 55

4.1 The Excluded Volume Model for the at sensor . . . . . . . . 574.1.1 Small potentials . . . . . . . . . . . . . . . . . . . . . 604.1.2 Large potentials . . . . . . . . . . . . . . . . . . . . . 654.1.3 Comparing the small and the large potential regime . 68

4.2 The Excluded Volume Model for the cylindrical sensor . . . . 694.2.1 Small potentials . . . . . . . . . . . . . . . . . . . . . 714.2.2 Large potentials . . . . . . . . . . . . . . . . . . . . . 72

4.3 Comparing the excluded volume e�ects in the at and thecylindrical geometry . . . . . . . . . . . . . . . . . . . . . . . 72

5 The Partially Excluded Volume Model 77

5.1 The partially Excluded Volume Model for the at sensor . . . 785.2 The Partially Excluded Volume Model for the cylindrical sensor 895.3 Comparing the e�ects of a partially excluded surface layer for

the at and the cylindrical geometry . . . . . . . . . . . . . . 90

6 Discussion and Conclusion 93

Chapter 1

Introduction to nanowire

biosensing

During the last half century, a dramatic downscaling of electronics has takenplace [1]. In 1965 Gordon Moore predicted that the number of transistorson a computer chip would double every two years. This is commonly knownas Moores Law. So far Mr. Moore has been right. The downscaling has ledto many technological innovations, from supercomputers to pocketsized mo-bile phones. A new technological opportunity for healthcare can be o�eredby miniaturized biosensors which use nanowires for sensing, and have po-tential for on-chip integration [2]. Due to the small dimensions, nanowireshave a large surface-to-volume ratio, making them highly sensitive. Dueto the high sensitivity, the nanowire biosensor can be used to detect DNAconcentrations in biological samples, without having to increase the DNAconcentration with time consuming PCR (Polymerase Chain Reactions).Biosensors based on silicon nanowires provide fast, simple, label-free electri-cal detection of biomolecules. Hundreds of diseases are diagnosable by themolecular analysis of nucleic acids, and the number is daily increasing [3].

Philips saw the opportunity for a direct electrical readout with a fastand in-expensive analysis of samples. The group I worked in at Philipsworks on nanowire sensing, which has applications ranging from air qualityto DNA testing. This thesis is written for this project of Philips Researchand focuses on biosensing.

The goal in nanowire biosensing is to use a nanowire �eld-e�ect transistor(FET) to measure what the concentration is of a certain biological moleculein the sample. The working principle of a FET is described in chapter 2.The basic idea of the nanowire sensor is the following (see �gure 1.1). Thenanowire surface is treated such that the biological molecule of interest canattach to the surface, while other molecules can not. When this is the casethe sensor is said to have a high selectivity. Depending on the concentrationof biomolecules in the sample, a certain number of molecules will attach

5

6 INTRODUCTION TO NANOWIRE BIOSENSING

to the sensor surface. The presence of the biomolecules close to the sensorsurface changes the electrostatic environment of the sensor. This causes achange in the surface potential, which is what the nanowire FET translatesinto an electrical signal.

Figure 1.1: Schematic of nanoscale biosensor. Biomolecules in solution can attachto the sensor surface. The presence of the biomolecules close to the sensor surfaceis detected by the Field-E�ect based biosensor. Source: [2]

The change in electrostatic environment of the sensor can be causedby di�erent properties of the biomolecule. The most commonly acceptedin uence that the biomolecule has on the electrostatic environment of thesensor is due to its charge. The e�ect of the charge is strongly in uencedby the fact that the solution in which the biomolecules are dissolved usuallycontains ions.

The ions in the system screen a part of the biomolecule charge, and thedetection of biomolecules becomes more di�cult. Therefore it is generallysuggested that the salt concentration should be kept low. This is howevernot always possible since biomolecules may deform, or are not able to attachto the sensor surface when the conditions deviate too much from bodilyconditions.

Estimations of the sensor signal due to the biomolecule charge whichtake the e�ect of ions into account predict less sensitivity than what hasbeen shown experimentally [3]. The source of this experimentally observedsignal generation is not well understood [3] [2], and it is suggested [3] thatnot only the charge of the biomolecules, but also the volume may have anin uence on the electrostatic surrounding of the sensor.

The goal of this thesis is to improve the understanding of the detectionmechanism of the nanowire biosensor. Therefore not only the e�ect of thecharge of the biomolecules is studied, but also the e�ect of the volume, while

INTRODUCTION TO NANOWIRE BIOSENSING 7

focusing on the e�ect of the ions in the biological solution.The volume of the biomolecule has two e�ects on the electrostatic envi-

ronment. First, the dielectric constant will become di�erent in a thin layeron the sensor surface. Second, ions in the vicinity of the sensor surface willbe redistributed.

To understand whether the behavior of the nanowire sensor is related toits geometry or that it is a general feature of FETs when used for biosensing,both the e�ects of charge and of volume have been studied for both thecylindrical nanowire FET and the at FET. For ease of calculations thenanowire FET is considered to be an in�nitely long cylinder, and the atFET an in�nite plane. There was no back gate included in the model,and the FETs are considered to be metallic. The biomolecules were not allincluded in the model separately, but modelled as a uniform layer. Since thediameter of the nanowires that are fabricated with top-down techniques isin the order of tens of nanometers, no quantum e�ects need to be includedin the models.

In chapter 3 only the in uence of the charge of the biomolecules is stud-ied. In chapters 4, and 5 the e�ect of volume is studied. In chapter 4 resultsare presented for a dense layer of biomolecules on the sensor surface, whilein chapter 5 the surface layer is only partially occupied by biomolecules.Whenever possible analytical expressions are given, in cases where this wasnot possible the results have been calculated with a �nite element computersimulation. In chapter 6 the conclusions are presented, showing the di�er-ence between the in uence of the charge and the volume of biomolecules,and the di�erence between the two geometries.

8 INTRODUCTION TO NANOWIRE BIOSENSING

Chapter 2

Basics of biosensing with

FETs

The physical mechanism underlying sensing remains controversial, as dis-cussed in the introduction. Previously suggested mechanisms are electro-static gating, changes in gate coupling, carrier mobility changes, and Schot-tky barrier e�ects [4]. Heller et al. [4] have shown that sensing is dominatedby electrostatic gating. In this thesis it is studied which properties of thebiomolecule and the surrounding electrolyte have the largest e�ect on thiselectrostatic gating.

When performing a biosensing experiment one wants to measure an elec-trical signal and thereby �nd the concentration of a certain biomolecule inthe biological sample. This can be done with the use of a �eld-e�ect tran-sistor or FET.

The response of FETs to an applied surface voltage is well known [5].However, in the case where the FET is used for biosensing, the surfacepotential is not �xed by a gate, but is both dependent on the environmentof the sensor in which the biomolecules and ions are present, and on thecharacteristics of the FET (section 2.1).

The theoretical description of biosensing, can be divided into two parts.The �rst part contains the physics of the FET device, and the second partcontains the physics in the biological solution. The boundary between thesetwo parts is at the sensor surface. The potential on this boundary is depen-dent on both parts. In this thesis it was chosen to focus on the second part,the biological solution. For the FET a highly simpli�ed model is taken.

For the description of the physics in the biological solution the proper-ties of the biomolecules, such as the attachment to the sensor surface, thecharge, and the volume are necessary. These properties vary greatly frombiomolecule to biomolecule, and are also dependent on the physical param-eters of the environment such as temperature, local potential and local ionconcentration. Here it was chosen to focus on the e�ect of the electrolyte

9

10 BASICS OF BIOSENSING WITH FETS

on the sensing mechanism. The properties of the biomolecules themselvesare not studied. Rough estimations of the value of the charge, the volume,and the number of biomolecules that attach to the sensor surface are madein section 2.2.

As was mentioned in the introduction, the sensing mechanism is studiedboth for a cylindrical device and for a at device. The di�erence in currentresponse of the two di�erent FET geometries to biomolecule attachment iscaused by several factors.

First of all di�erent geometries can give di�erent signals because thesignal change upon attachment depends on the signal before attachment,and this depends on the area of cross section and the material of the FETdevice. To eliminate the e�ects of device size and the speci�c material used,it is decided to discuss the results of the calculations in terms of the chargeinduced in the FET per unit length, instead of the current change. This isdiscussed in section 2.3.

The second factor is that the amount of charge induced on the FETdue to a certain surface potential is di�erent, since the two geometries havedi�erent capacitances. The induced charge in the FET can be calculatedas a function of the surface potential, expressed in terms of its capacitance.The capacitance of semiconductor devices can be very complex. Since thecapacitance is not the main topic of interest, simple capacitances have beenchosen assuming the FETs are metallic.

Thirdly, the potential on the FET surface is di�erent in the di�erentgeometries. It depends not only on the charge of the biomolecules, but alsoon the capacitance of the FET and the charge in the ionic double layersurrounding the biomolecules and the sensor. Both this capacitance andthe charge induced in the double layer are di�erent for the two geometries(section 2.4 and 2.5).

The theory discussed in the remainder of this chapter will be combinedto do calculations on di�erent sensing mechanism models in chapters 4 to 5.

2.1 The �eld-e�ect transistor

The biosensor discussed in this thesis is based on a �eld-e�ect transistor(FET). A simple representation of a FET is given in �gure 2.1. The topgate is used to �x the potential on the top of the FET. The conductivity ofthe semiconductor material underneath the top gate changes as a functionof this surface potential.

A semiconductor is characterized by the fact that the fermi level is be-tween the conduction and the valence band. Depending on the circumstancessome electrons in the semiconductor material will have enough energy tooccupy a state in the conduction band. This number of electrons in theconduction band determines the conductivity of the material. By changing

BASICS OF BIOSENSING WITH FETS 11

Figure 2.1: In a �eld-e�ect transistor the conductivity of the path from source todrain is in uenced by the potential of the top gate.

the top gate potential the energy of the conduction band will change, whichincreases or decreases the number of electrons in the conduction band (�gure2.2). The conductivity in the path from source to drain is therefore directlydependent on the gate potential. Measuring the electrical signal will tell uswhat surface potential is applied. The relation between the surface poten-tial and the electrical signal does not have to be a simple function, but canbe very complex, and it also depends on the geometry, the material of thedevice, and the dopant molecules in the material.

Figure 2.2: Band diagram for a FET. In the center an external voltage is appliedthat causes bending of the bands.

In a chemical �eld-e�ect transistor, or ChemFET, the conductivity ofthe device is also dependent on the surface voltage. Only now the surfacepotential is not in uenced by a top gate, but by molecules on the sensorsurface, and the FET can therefore be used as a chemical sensor. When themolecules on the surface participate in a chemical process, the electrostaticsurrounding of the sensor surface is changed (for example due to a change


in surface charge), which changes the potential. The same principle can beused for biosensing, where the presence of the biomolecules in uences theconductivity of the device (see �gure 2.3).

Figure 2.3: Schematic cross-section of a ChemFET. The presence of the moleculesin uences the surface potential. The change in surface potential is translated bythe FET into an electrical signal.

The Nanowire FET sensor is similar to the ChemFET, except thatthe sensitive semiconducting layer of the ChemFET corresponds to a thinnanowire in the nanowire FET (see �gure 2.4). It is important that thenanowire surface is functionalized with highly speci�c receptors, or capturemolecules, such that only the speci�c molecules of interest will be able toattach to the sensor surface and cause a signal. Only then will a measure-ment of the current provide information on the presence of the biomoleculeof interest. It was mentioned before, that the relation between signal andsurface potential can be rather complex. Depending on the value of thesurface potential, a small change in the potential will result in a large or asmall signal. The back gate of the device can be used to tune the FET suchthat the conductivity of the wire changes fast with a small change in thesurface potential.

a.) b.)

Figure 2.4: (a.) Cross-section of the nanowire sensor along the nanowire axis.(b.) Cross-section of the nanowire sensor perpendicular to the nanowire axis.


In many sensing experiments nanowire FETs are used instead of atFETs, since the nanowire FET is more sensitive than the at FET. Someof the bene�ts of the nanowire FET are the high surface-to-volume ratio,the fast di�usion of biomolecules to the sensor surface [6], and the fact thatonly a few attached biomolecules have a large in uence on the conductance,also referred to as pinching. These characteristics are a good reason to usenanowires for sensing experiments.

The sensing experiments that are being performed by Philips Researchare therefore also done using nanowire FETs. In �gure 2.6 a SEM pictureis shown of a nanowire device. For a large scale production the use of topdown fabrication techniques of the nanowires is required. These fabricationtechniques set a lower limit on the diameter of the nanowires of about 30nm. A lower limit of about 5 nm on the thickness of the isolating oxide isgiven by the fact that below this thickness the layer is no longer isolating.The length of the wires can be varied, and is usually taken to be in theorder of 1 �m. For ease of calculations the nanowire �eld-e�ect transistordescribed above is simpli�ed. The nanowire is considered to be an in�nitelylong cylinder with a diameter of 2b = 30nm, covered with an oxide layerwith thickness tOX = 5nm, suspended in the biological solution (see �gure2.5). The in uence of the back gate is not taken into account. As mentionedbefore the device is assumed to be metallic.

Figure 2.5: Cross-section of the nanowire FET sensor showing the nanowire, theoxide shell, and the receptors, some with a biomolecule attached, surrounded bythe electrolyte solution. Source: [2], �gure edited.

2.2 Biomolecule properties

To understand how the attachment of biomolecules in uences the electro-static environment of the biosensor, some properties of the biomoleculeshave to be known.

First of all we need to know the relation between the biomolecule concen-


Figure 2.6: A SEM (Scanning Electron Microscopy) picture of a nanowire.

tration in solution and the surface density of the attached biomolecules onthe sensor surface, this will be discussed in section 2.2.1. This is dependenton many factors such as the ion concentration in the electrolyte solution,the local potential, and the temperature.

Second, we need to know what the properties of the biomolecules arethat can change the electrostatic environment of the sensor (sections 2.2.2,2.2.3). In these sections rough estimations of the above parameters will bemade.

2.2.1 Biomolecule attachment

The number of biomolecules that attach to the surface depends on a numberof factors. One important parameter is the amount of sites on the surface thebiomolecule can attach to. This is often referred to as the number of capturemolecules on the surface, while the biomolecules of interest are referred toas the target molecules. The density of capture molecules should not be toolarge, since neighboring capture molecules should not hinder one another tocapture a target molecule. This density should be in the range from about 2�1013 to 1:3�1012 molecules cm�2 [3]. The number of biomolecules that attachto the sensor surface also depends on the concentration of biomoleculesin the sample solution, and the capture and dissociation constants of thebinding. In reality these constants and thus the amount of molecules thatattach to the surface, are dependent on factors, such as the local potentialand the temperature [7]. However, in this thesis a rough estimation of thebiomolecule attachment is used and the e�ect of the above factors is nottaken into account. Time dependence is not taken into account, only thesteady state situation is considered here.

To estimate the order of magnitude of the biomolecule attachment aDNA molecule is used. A rough estimation of the number of attached


biomolecules in the case of DNA molecules is given by Alam and Nair [2].For a capture molecule density of 1 � 1013 cm�2 and an available biomoleculedensity of 1 � 10�9M, the number of attached biomolecules on the surface is0:23 � 1013 cm�2.

To visualize the degree of biomolecule attachment the density of captureand target molecules on the sensor surface is shown in �gure 2.7. In this�gure the capture molecule density is 1:25�1013 cm�2, and the surface densityof target molecules, N , is 0:31 � 1013cm�2. The circles are 2 nm in diameter,which corresponds to the diameter of a DNA helix. Throughout this thesisthe number of attached biomolecules is taken to be 0:3 �1013cm�2. It shouldbe considered to be an upper limit of the attached biomolecule density, sincein reality we are interested in biomolecule concentrations down to 10�15M(femtomolar).

Figure 2.7: An image showing the density of capture molecules on the sensorsurface (white circles, 1:25 �1013 cm�2), and the density of biomolecules (red circles,0:31 � 1013 cm�2).

The density of biomolecules on the sensor surface is necessary to estimatethe e�ect on the FET, but for this also the properties of the biomolecules,such as the charge and the volume, is necessary. This will be considered inthe following sections.

2.2.2 Surface Charge

To know the order of magnitude of the biomolecule charge that in uencesthe conductivity of the FET, again a DNA molecule is taken as an example.

A picture of a DNA helix is shown in �gure 2.8. The phosphate backboneof DNA is charged. Each base contains one electron charge. Depending onthe length of the DNA fragment under consideration, the biomolecule willhave a di�erent charge. However, due to screening of the ions in the elec-


Figure 2.8: A DNA helix oriented perpendicular to the sensor surface. The phos-phate backbone is negatively charged. The Debye length for an ion concentrationof 0.1 M is shown on the right (picture taken from [3]).

trolyte solution, only the charge that is close enough to the sensor surfacewill have an in uence on the FET. The Debye length gives the length scaleover which the �elds of the charges in the electrolyte are screened. If acharge is farther away than this Debye length it can be considered to haveno in uence. The ion concentration of the electrolyte solution must have anion concentration that is close to physiological solutions. Otherwise the twonegatively charged DNA strands will repel each other and the target DNAstrand will not be able to attach. Taking an ion concentration of 0.1 M, we�nd a Debye length of about 1 nm. Within a distance from the surface of 1nm, 3 DNA bases are present if the DNA helix is oriented perpendicular tothe surface, and more if the helix lies at on the surface. Each base has acharge of one electron charge, e. In an ionic solution with monovalent ionshowever, counterions compensate for 74 % of this charge [8]. To summa-rize: every DNA molecule has 3 bases that must be taken into account inthe model with a charge of 0.26 e. This leads to the estimation that eachbiomolecule has a charge of about 1 electron charge, which corresponds to


the value used in literature [2].

Taking the surface density of the attached biomolecules as was givenin section 2.2.1, we obtain a charge density of attached biomolecules of0:3 � 1013C=(e cm2), or 4:806 � 10�3C=m2. Note that DNA is a stronglycharged biomolecule, so for other biomolecules, this could be much less.

In addition to the biomolecule charge, the oxide on the FET also containscharge. At the Philips lab oxide charges typically range from 1010e cm�2

to 1012e cm�2. These values correspond to the typical oxide charges asgiven in textbooks [9]. Assuming a high quality oxide preparation the oxidecharge can be considered to be 1010e cm�2. This is more than two orders ofmagnitude smaller than the surface charge density of the biomolecule chargethat was given above, and can be neglected. However, when using an oxidelayer of lower quality, or when there is less biomolecule charge in the system,the oxide charge must be included.

In this thesis the biomolecule charge is modelled as a uniform surfacecharge. The theory described is therefore not applicable to single moleculedetection.

2.2.3 Volume

To make an estimation of the order of magnitude of the volume of thebiomolecule, DNA is taken as an example again. The DNA molecule can bemodelled as a rod [10] with diameter of 2 nm. For a sequence of 30 basesthe length is about 10 nm [3].

The attachment of the biomolecules changes the properties of the envi-ronment of the sensor only in a thin layer. The thickness of this layer isdependent on the orientation of the biomolecules. If they are oriented per-pendicular to the sensor surface the thickness of the altered layer is largest,in the case described above this is 10 nm. The total fraction of the volumethat is occupied by the biomolecules is small in this layer. If the biomoleculesare oriented parallel to the sensor surface the in uenced layer is thin, downto 2 nm, but the fraction of the volume in this layer that is occupied bythe biomolecules is larger. This volume fraction that is occupied by thebiomolecules, from hereon called the excluded volume fraction, is calculatedbelow for these two extremes.

In �gure 2.9 (a.) the biomolecules are oriented perpendicular to thesensor surface. Taking the density of biomolecules as given in section 2.2.1we obtain a total excluded volume fraction of f = N�rDNA

2 = 0:094 � 0:1in a layer with a thickness of 10 nm, where N is the number of attachedbiomolecules per cm2, and rDNA the radius of the DNA helix.

In �gure 2.9 (b.) the biomolecules are completely at on the surface, ina layer with a thickness of 2 nm. The total excluded volume fraction in thiscase yields f = N�rDNA

2l=tbio = 0:47 � 0:5, with l the length of the DNAsegment, above said to be 10 nm, and tbio the thickness of the layer around


the sensor in which the biomolecules are present, here 2 nm. This equationcan also be used when the DNA molecules have an orientation in betweenperpendicular and at, such that tbio has a value between 2 and 10 nm.

DNA has a rather elongated shape. For more round molecules the dif-ference in excluded volume of the two orientations will not be this clear, andthe excluded volume fraction and the layer thickness will be in between thetwo above results.

a.)

b.)

Figure 2.9: A DNA helix can be modelled as a rod with a diameter of about 2nm. (a.) The biomolecules are oriented perpendicular to the sensor surface. (b.)The biomolecules lie at on the surface.

For a cylindrical sensor, the excluded volume fraction changes as a func-tion of distance from the surface. This will only be noticeable if the layerthickness is rather large compared to the cylinder radius. The excluded vol-ume fraction then becomes f(r; d) = N�RDNA

2lb=(tbior), with b the outerradius of the nanowire sensor.

The excluded volume fraction indicates what fraction of the electrolytesolution is pushed away around the sensor surface. This has two e�ects. The�rst is that ions are excluded from this region, which changes the chargedistribution in the vicinity of the sensor, and thereby decreases the e�ectof screening. The second e�ect is that the dielectric constant is di�erent aswill be explained in the next section.

Dielectric constant

After attachment of biomolecules the dielectric constant in a thin layer ontop of the surface of the sensor has a di�erent value than before attachment.This is caused by various factors. First of all, the dielectric constant of thebiomolecules di�ers from that of the surrounding electrolyte solution. So ef-fectively, the dielectric constant around the sensor changes upon biomoleculeattachment. In this section a rough estimation will be made of this e�ect.Other e�ects, such as reduced intermolecular hydrogen bonding of the wa-ter near the macromolecular surface and the high counterion concentrationsaround the molecules [10], are neglected.


A part of the volume around the sensor has a dielectric constant ofthe biomolecules, �r;bio, in the rest of the volume the dielectric constant isequal to that of the electrolyte solution, �r;w. The combined material canbe described as a uniform layer of an electrostatically equivalent materialwith a new e�ective dielectric constant. The problem addressed in thissection is what value to take for this e�ective, new dielectric constant, �r;new.Assuming the biomolecules have a rodlike shape, two di�erent geometriescan be considered. One in which the biomolecules are assumed to be orientedperpendicular to the sensor surface, and one in which they are orientedparallel to the sensor surface (see �gure 2.9). This gives rise to regions withdielectric constants as shown in �gure 2.10.

a.)

b.)

c.)

Figure 2.10: Schematic of a system consisting of regions with di�erent dielectricconstants (a.) and (b.), and an equivalent system with an e�ective uniform dielectricconstant that yields an electrostatically equivalent system.

If one assumes that the biomolecules are oriented perpendicular to thesensor surface the new dielectric constant is given by

�1r;new = f�r;bio + (1� f)�r;w; (2.1)

If the biomolecules are at on the surface we obtain

1

�2r;new=

f

�r;bio+(1� f)

�r;w: (2.2)

with f the excluded volume fraction as discussed above, and �r;bio the relativedielectric constant of the biomolecule, which is about 2 to 4 for DNA. Theseresults are shown in �gure 2.11. It can be seen that the dielectric constant is


changed more upon attachment of biomolecules when the biomolecules areoriented perpendicular to the sensor surface than when they are at on thesurface.

Figure 2.11:

2.3 Sensor sensitivity

A simpli�ed expression for the current through a material due to an applied�eld is given by [5]

I = vdriftq ncA; (2.3)

where vdrift = �E is the velocity of the charge carriers with � the mobilityof the carriers, and E the electric �eld, q is the charge of the charge carriers,nc is the number of carriers per unit volume, and A is the area of crosssection through the device perpendicular to the direction of current ow.

The current after adsorption of biomolecules is di�erent due to the in-duced charge per unit length in the FET, ��FET,

I = vdrift q ncA+ vdrift��FET: (2.4)

In this simple model it is assumed that �, and therefore vdrift, does notdepend on the carrier density. The sensitivity of the sensor is given by [2]

�I

I=

��FETe ncA

: (2.5)

The above expression shows that the sensitivity of the sensor is stronglydependent on the material chosen via the density of charge carriers, nc, andon the geometry and size through the size of the cross section, A.

The induced charge in the device, ��FET, is related to the active surfacearea of the device, which is, in this case, the total surface of the device


which is 2�r. The area of cross section �r2 of a nanowire FET is smallcompared to its surface, since the diameter of the nanowire is so small.Often the sensitivity of the nanowire sensor is ascribed to this property,referred to as the high surface-to-volume ratio. A high surface-to-volumeratio, however, can also be achieved in the at geometry by making thinnanoribbons. Therefore this is not considered to be a characteristic propertyof a nanowire in this thesis.

For the at and the cylindrical FET that are compared in this thesisthe material is assumed to be the same, and the device geometry is chosensuch that the surface areas and the areas of cross section are equal. Thisrelates the radius of the cylindrical device to the width and thickness of the at device. The width of the at device, W , must be 2�(a + tox), and itsthickness, d, must be d = �a2=(2�(a+ tox)) (see �gure 2.12).

To remove the e�ect of the material used, the results will be discussed interms of the charge induced in the FETs per unit length, �FET. Assumingthe active surface area, the carrier density, and the cross sectional area to bethe same in the di�erent situations, the value of the induced charge per unitlength, �FET, can be used directly for comparison of the sensitivity of thedi�erent model systems. This way the e�ect of the material and the size isremoved, and the e�ect of presence of the biomolecules and the electrolytecan be studied.

Also the attached biomolecule density on the device surface due tobiomolecule absorption is assumed to be the same in both geometries. Thisgives us two devices with the same amount of biomolecules per unit length onthe surface. Under these assumptions the di�erence between the responsesof the two geometries to the same amount of biomolecules can be studied.

2.4 Capacitance

The amount of charge that is induced in the FET due to a certain sur-face potential depends on the capacitance of the device. In this section anexpression for the capacitance of a at and of a cylindrical FET is derived.

The capacitance per unit area of a capacitor is

C =�

V; (2.6)

with � the charge per unit area on each capacitor plate, and V the potentialdi�erence across the two plates.

In the case of the FET the capacitance of the device can be taken as

C =�FET

(VS � VFET); (2.7)

where VS is the potential on the surface of the FET and VFET the potentialin the FET. This capacitance is di�erent for di�erent device geometries. The


a.)

b.)

Figure 2.12: A cross section of a at and a cylindrical FET. The direction ofcurrent ow is perpendicular to the paper. To be able to make a sensible comparisonbetween the at and the cylindrical FET, the geometries must be chosen such thatthe active surface areas of the FETs, and the areas of cross section (the orangeregion) are the same. This implies W = 2�(a+ tox), and d = �a2=(2�(a+ tox)).

capacitance of the device gives the charge per unit area induced in the FETfor a certain surface potential,

�FET = C(Vsurface � VFET): (2.8)

When the FET is grounded and the surface potential is V0 the inducedcharge per unit area is equal to

�FET = CV0: (2.9)

If the surface potential is positive, a negative charge will be induced in theFET, and vice versa. Since charges of opposite sign will be induced in theFET, the capacitance is a negative quantity. It is seen that for a largecapacitance, a small change of the potential will induce a large change inthe charge in the FET. Therefor a large capacitance is bene�cial for sensitivesensing.

The capacitance of a semiconductor FET can be dynamic and very com-plex. Since the object of study here is the e�ect of the electrolyte, a simple


expression is taken for the capacitance, such that all e�ects can be ascribedto the behavior of the electrolyte.

The device will be considered to be metallic, since this yields a sim-ple expression for the capacitance. This assumption is only valid for thesemiconductor FET for a certain range of the surface potential. For non-metallic devices the simple expression of the metallic capacitance that wasused throughout this thesis must be replaced by the real, more complex,semiconductor expression. The calculation itself remains the same.

2.4.1 The capacitance of a at metallic FET

The capacitance of the at geometry can be calculated by considering aGaussian pillbox containing a part of the metal-oxide interface. This yieldsthe value of the electric �eld in the oxide layer. The potential di�erencebetween the conducting layer and the oxide surface can then be calculatedby integrating the �eld over the oxide thickness. Fringing of the electric �eldat the edges is ignored. The capacitance per unit area for the at geometryis

C at = ��0�r;oxtox

: (2.10)

2.4.2 The capacitance of a cylindrical metallic FET

The capacitance of the cylindrical capacitor can be calculated by consideringa Gaussian surface in the shape of a cylinder in the oxide enclosing the metalcore of the cylindrical FET. The result for the capacitance per unit area ofthe cylindrical geometry is equal to

Ccyl = � �0�r;ox

a ln(1 + toxa )

: (2.11)

2.4.3 The capacitances of a at and a cylindrical metallic

FET compared

To be able to compare the capacitances of the two di�erent geometries, thecapacitance per unit length is considered for both devices. As discussedbefore the width of the at device must be 2�b, where b = a + tox. Thecapacitance per unit length of the at FET is equal to

~C at = �2�b�0�r;oxtox

= �2��0�r;ox a

tox

�1 +

toxa

�: (2.12)


The capacitance per unit length of the cylindrical FET is

~Ccyl = � 2��0�r;ox

ln(1 + toxa )

: (2.13)

The capacitance of the at device is bigger than that of the cylindricaldevice. This indicates that for the same surface potential more charge isinduced in the at FET than in the cylindrical FET, indicating that the at FET is more sensitive to small changes in the surface potential. In�gure 2.13 the capacitances are plotted as a function of tox=a. It can beseen that for a smaller tox=a, more charge is induced on the FET devicesper unit length. The sensitivity of the FET device to a surface potentialtherefore increases for smaller oxide thicknesses. In the limit tox=a� 1 thetwo capacitances are the same, Ccyl � C at.

a.)

b.)

Figure 2.13: (a.) The capacitances of the at (green line) and the cylindrical(red, dashed line) metallic FET per unit length of device. In the limit tox=a ! 0the two capacitances are equal as can be seen in (b.) where the ratio ~Ccyl= ~C at isplotted. �ox = 4.


2.5 Screening

To ensure proper attachment of the biological target molecules to the capturemolecules, biosensing experiments are often performed under physiologicalconditions. The solution containing the biomolecules therefore has a largeion concentration, which has to be taken into account. The ions in theelectrolyte form double layers which screen the electric �elds in the system.Depending on the potential in the system, the ions will have a certain dis-tribution, and the other way around, the potential itself is dependent on thedistribution of the ions. This means that additional charge of the doublelayer also has an in uence on the potential at the FET surface.

In the following derivations it is assumed that the electric potential in theelectrolyte region obeys the Poisson-Boltzmann equation. For a grounded1-1 electrolyte solution this is (in SI units)

r2V (r) = � eI0�0�r;w

�exp

�� eV

kBT

�� exp

�+

eV

kBT

��

= +2eI0�0�r;w

sinh

�eV

kBT

�; (2.14)

where e is the electron charge, I0 is the concentration of both the positive andthe negative ions in the electrolyte, �r;w is the relative dielectric constant ofthe electrolyte solution, kB is the Boltzmann constant, and T is the absolutetemperature. This equation can be used to �nd the total charge containedin the double layer per unit area of the device.

The Boltzmann statistics in this equation do not take into account thevolume of the ions, since the ions in the solution are considered to be pointcharges. In regions with a high potential the above equation can predict anion concentration exceeding the concentration maximally possible for hardspheres with a radius of a couple of�angstr�oms, which is the ion radius. For a1-1 monovalent electrolyte solution of 0.1 M of ions with a size of 2 �A, the ionconcentration exceeds the maximum hard sphere concentration at a voltageof 150 mV, which corresponds to a dimensionless potential = eV=(kBT ) ofabout 6. In this thesis, however, most of the time such a potential will not bereached, therefore the Poisson-Boltzmann equation is assumed to be validhere. When higher potentials are reached a modi�ed Poisson-Boltzmannequation has to be used [11].

In the following sections the charge contained in the double layer iscalculated as a function of the interface potential V0 both for a at and acylindrical geometry.


2.5.1 Screening in the at geometry

In this section the charge in the double layer that is formed on top of a atpotential is calculated. The system is shown in �gure 2.14.

Figure 2.14: Screening in the at geometry. The potential is zero in the electrolytefar from the �xed potential surface V0.

The 1 dimensional version of the Poisson-Boltzmann equation for agrounded 1-1 electrolyte is (see equation (2.14)):

d2(x)

dx2= �2 sinh ((x)) ; (2.15)

where (x) = eV (x)=(kBT ) is the dimensionless potential, x is the distanceto the surface where the potential is 0 = eV0=(kBT ) (see �gure 2.14), and� is the inverse of the Debye length of the solution. The Debye length is thescale over which the ions in the solution screen electric �elds. The expressionfor � is

�2 =2e2I0

�0�r;wkBT; (2.16)

with e the electron charge, I0 the ion concentration in the bulk of the elec-trolyte, and �r;w the relative dielectric constant of the electrolyte.

Rewriting the left hand side of the Poisson-Boltzmann equation, andintegrating once over the potential yields

Z (1)

(x)

d2(x)

dx2d =

Z (1)

(x)

1

2

d

d

�d

dx

�2d

= �12

�d(x)

dx

�2: (2.17)


Integrating the right hand side over the potential yieldsZ (1)

(x)

�2 sinh ((x)) d = �2�2 sinh2�(x)

2

�; (2.18)

where for the derivation of the two above equations we used the fact thatd=dx = 0 far from the surface due to screening by the ions, and that thesolution is grounded so (1) = 0.

Combining equations (2.17) and (2.18), we obtain

d

dx= �2� sinh

�(x)

2

�: (2.19)

From Poisson's equation we know:Z1

x

d2V (x)

dx2dx = �dV (x)

dx

=

Z1

x

�(x)

�0�r;wdx; (2.20)

andR1

0 �(x)dx = �DL.This gives the result for the total charge of the ions in the double layer

per unit area of the potential interface [12]:

� atDL =�2�0�r;wkBT�

esinh

�0

2

�

= �2p2�0�r;wkBTI0 sinh

�eV02kBT

�: (2.21)

The second line shows the dependence on all physical parameters. Fromhereon we will keep on working with 0 and �, although both contain vari-ables such as T and �r;w. Therefore the dependence on temperature anddielectric constant is no longer manifest. It can be argued however thatfor biological applications the temperature and the dielectric constant ofthe solution are constants instead of variables, since the biological systemsimply imposes a value. The conformation of the biomolecules and possiblereactions with other molecules depend strongly on the surrounding medium.In systems that deviate too much from bodily conditions biomolecules candeform, or even disintegrate.

In the above equation the sign was chosen to be negative since for thecharges with an opposite sign of the potential it is energetically more favor-able to be in the region with that potential, than to be in the bulk of thesolution where the potential is zero.

The dependence on 0 is illustrated in �gure 2.15. It is seen that for� 1 the following approximation is valid

� atDL =��0�r;wkBT�0

e; (2.22)


Figure 2.15: The charge in the double layer per unit area of at interface potentialas a function of the dimensionless interface potential (equation 2.21), and the resultof the approximations 0 � 1 and 0 � 1. The values of the other variables arestandard (see table at the end of this chapter).

showing that in this regime the charge of the double layer per unit area islinear in 0. For 0 � 1 the correct approximation is

� atDL =��0�r;wkBT�

eexp

�0

2

�; (2.23)

which has an exponential dependence on the interface potential. As dis-cussed before, the Poisson-Boltzmann theory is not valid for high values of0. Therefore this large potential approximation is only valid in a smallinterval of 0, and this approximation should be used with care.

2.5.2 Screening in the cylindrical geometry

In this section the charge in the double layer that is formed around a cylin-drical potential is calculated. The system is shown in �gure 2.16.

The cylindrical Poisson-Boltzmann equation for a grounded 1-1 elec-trolyte is:

d2

dr2+1

r

d

dr= �2 sinh((r)); (2.24)

with r the distance to the potential surface where 0 = eV0=(kBT ) (see�gure 2.16).

Rewriting this equation in dimensionless parameters

d2

d(�r)2+

1

�r

d

d(�r)= sinh((�r)); (2.25)

it is easily seen that for �R� 1 the second term on the left hand side dropsout since �r � �R and the at Poisson-Boltzmann equation is recovered.


Figure 2.16: Screening in the cylindrical geometry. The potential is zero in theelectrolyte far from the �xed potential surface V0.

The cylindrical Poisson-Boltzmann equation can not be solved analyt-ically. However, this equation has been solved by Ohshima [13] under theassumption that �R� 1, with R the radius of the cylindrical potential, and� the inverse screening length. The charge of the double layer per unit areaof the cylindrical potential is found to be:

�cylDL = �2�0�r;wkBTe

� sinh(0

2)

"1 +

�2 � 1

cosh2�04

�# 1

2

; (2.26)

where = K0(�R)=K1(�R), with Kn the modi�ed Bessel function ofthe second kind.

Figure 2.17: The charge in the double layer per unit area of cylindrical interfacepotential as a function of the dimensionless interface potential (equation 2.26), andthe result of the approximations 0 � 1 and 0 � 1. The black dots indicate theresults of the �nite element method. I0 = 0:1M, �r;w = 80,R = 15nm.


For 0 � 1 the above expression can be approximated by

�cylDL =��0�r;wkBT�0

e

�1 +

�2 � 1

1

� 12

=��0�r;wkBT�0

e �1; (2.27)

For 0 � 1 the following approximation is valid

�cylDL =��0�r;wkBT�

eexp(0=2); (2.28)

In �gure 2.17 the charge in the double layer is plotted together with theresult for the approximations 0 � 1 and 0 � 1. The black dots indicatethe results that were obtained using a �nite element computer calculation. Itis seen that the analytical results, that were derived using the approximation�R � 1 [13], agree well with the �nite element results. The above derivedanalytical expression (equation (2.26)) is therefore useful for the situationwe are considering here, �R = 15:5, and will be used throughout this thesis.As discussed before, the Poisson-Boltzmann equation is not valid for largevalues of the potential. Therefore this equation should be used only in acertain interval of 0.

2.5.3 Comparing the double layers of the at and the cylin-

drical geometries

The charges in the double layers of the two geometries per unit area of theinterface potential, � atDL and �cylDL, are compared most easily by taking theirratio (equation 2.21 and 2.26)

�cylDL

�flatDL

=

"1 +

�2 � 1

cosh2�04

�# 1

2

: (2.29)

When this expression equals 1 the double layers of the at and the cylin-drical geometry contain an equal amount of charge. This is the case when �1 goes to one, which corresponds to �R� 1 as can be seen in �gure 2.18.When �R � 1, the radius of the cylindrical potential is large compared tothe Debye screening length, ��1. It is an intuitive result that the expres-sions for the at and the cylindrical geometry are equal when the radius islarge compared to the only other length scale in the system, which is thescreening length. The above expression also equals 1 when the hyperboliccosine reaches high values, which is the case for 0 � 1.

The at and the cylindrical geometry give identical results for the chargein the double layer, except when 0 and �R are both small.

Figure 2.19 shows the analytical results of the charge in the double layerper unit area, �DL, as a function of 0, of the at geometry (equation (2.21)),


Figure 2.18: The function �1 = K1(�R)=K0(�R) plotted as a function of �R,with Kn the modi�ed Bessel function of the second kind, � the inverse Debyescreening length, and R the radius of the cylindrical potential.

and the cylindrical geometry (equation (2.26)), and it shows the results thatwere obtained for the cylindrical geometry using a �nite element method(COMSOL).

a.) b.)

c.)

Figure 2.19: The dependence of the charge in the double layer per unit area ofthe device, �DL, on the dimensionless potential 0 for two di�erent values of �R.The green solid line and the red dashed line are the analytically derived chargesin the double layer per unit area of a at and a cylindrical FET device geometryrespectively, � atDL and �cylDL. The black dots show the charge in the double layer perunit area of the cylindrical FET calculated with a �nite element method, �COMSOL

DL .(a.) To obtain a value of �R = 0:5 we chose I0 = 0:01M and R = 1:5 nm. (b.)The value �R = 15:5 is found when using the standard values (I0 = 0:1M andR = 15nm). (c.) The ratio of the analytical cylindrical and the at result for bothvalues of �R. In all �gures �r;w = 80.


In �gure 2.19 (a.) it can be seen that the three solutions of �DL are clearlydi�erent for �R = 0:5. The double layer of the cylindrical geometry containsmore charge per unit area of device than that of the at geometry. Figure2.19 (c.) is a plot of the ratio �cylDL=�

atDL showing this di�erence between the

analytical expressions in more detail. For small values of 0 the di�erenceis signi�cant, for larger values of 0 the plot tends to 1, indicating that the at and the cylindrical double layers contain the same amount of charge.

For this value of �R = 0:5 the assumption �R � 1, which was used forderivation of the cylindrical expression [13], does no longer hold. The resultsof the cylindrical geometry found using a �nite element method are alsoshown. It is seen that even in this regime the analytical calculations still givegood results. These results are however not important for biosensing. Toreach a value of �R = 0:5 a very small radius had to be chosen, R = 1:5 nm,and the ion concentration was not physiological, but I0 = 0:01M. Nanowirescan be made with such small diameters (e.g. carbon nanutubes), but notwith top down methods, which is necessary for large-scale production. Alsothe physiological ion concentration is necessary for correct binding. Figure2.19 (b.) therefore shows a more realistic picture where R = 15nm, andI0 = 0:1M. The ratios are again shown in �gure 2.19 (c.), where it is seenthat the di�erence between the two geometries is only a few percent forrealistic values of �R = 15:5.

In �gure 2.20 (a.) and (b.) the three solutions are shown as a functionof �R for two di�erent values of the potential, 0 = 2 and 0 = 6, respec-tively. The results of the at geometry and the results of the analytical and�nite element method for the cylindrical geometry look the same, since thedi�erence is only noticeable for very small values of �R. Figure 2.20 (c.) isa plot of the ratio of the at and the cylindrical analytical result for bothvalues of the potential, showing that only for small values of the potentialthe di�erence between the two geometries is substantial. The upper limit,and the lower limit are given by the black dashed lines. When 0 � 1equation 2.29 becomes

�cylDL

�flatDL

= �1: (2.30)

In �gure 2.20 (c.) the black dashed line shows this function as an upperlimit for the value of the ratio as a function of �R. When 0 � 1, the ratioof the charge in the double layer per unit area device in the cylindrical andthe at geometry equals one, which is also shown in the �gure.

It was seen that the analytical result of the cylindrical geometry, whichwas derived using the approximation �R � 1, gives correct predictions ofthe double layer charge for biosensing experiments. When comparing the at and the cylindrical geometry, it was seen that the double layer that formsaround a cylindrical potential contains more charge than that surrounding


a.) b.)

c.)

Figure 2.20: (a.) and (b.) The dependence of the charge in the double layer perunit area of the device, �DL, on �R for two di�erent values of 0. The analyticallyderived charge in the double layer per unit area of a FET device for the at ge-ometry, � atDL (green solid line) and the cylindrical geometry, �cylDL (red dashed line),and the charge in the double layer per unit area of the cylindrical FET calculatedwith a �nite element method, �COMSOL

DL (black dots) are shown. (c.) The ratio ofthe analytical cylindrical and the at result for both values of 0. In all �gures�r;w = 80.

a at potential. Noting however that for realistic nanowire diameters andion concentrations the value of �R is 15.5, and that the realistic results arelocated on the right of the graphs in �gure 2.20, it is concluded that forthe relevant regime for biosensing, the double layers of the two geometriescontain the same amount of charge.


Table 2.1: Overview of the estimations of the physical variables that were madein the previous sections.

ion concentration 0.1 M

relative dielectric constant of the ionic solution,�r;w 80

relative dielectric constant of the oxide, �r;ox 4

relative dielectric constant of biomolecules, �r;bio 2

outer nanowire radius, b 15 nm

inner nanowire radius, a 10 nm

oxide thickness, tox 5 nm

density of attached biomolecules, N 0:3 � 1013cm�2e�ective charge of biomolecule 1 e

temperature, T 293 K

Chapter 3

The Surface Charge Model

In this chapter the charge that is induced in the FET by the presenceof the attached biomolecules is discussed. The presence of the attachedbiomolecules is modelled as a surface charge, �S (see �gure 3.1). Thischarge is placed directly on the surface of the oxide layer. The value of thebiomolecule surface charge is chosen such that the e�ect of counterions is in-cluded (section 2.2.2). A simpli�ed, cylindrical model of the nanowire sensoris considered. The double layer charge is modelled as a surface charge, �DL,and will be described with Poisson-Boltzmann theory (section 2.5). Theinduced charge in the nanowire, �NW, which is the quantity of interest, isdescribed as an in�nitely thin, uniform layer of charge just below the surfaceof the nanowire (see �gure 3.2). Since the biomolecules are modelled as asurface charge this chapter will be called the Surface Charge Model (SCM).It is based on the work of Alam and Nair [2].

Figure 3.1: Left: A biomolecule contains multiple charges. Due to screeningonly the charges close to the sensor surface are sensed by the FET. Right: Thebiomolecules attached to the sensor surface are modelled as a surface charge withthe value of only those charges of the biomolecules that are closer to the sensorsurface than a couple of Debye screening lengths.

Using the theory discussed in the previous chapter, all ingredients areavailable to calculate the amount of charge induced in a FET when thebiomolecule surface charge is given. The only relation missing to solve theSurface Charge Model is the fact that all the charges in the system addedtogether, should yield zero charge:

35

36 THE SURFACE CHARGE MODEL

�FET + �S + �DL = 0; (3.1)

with � the charge per unit length of FET sensor. This means that all thebiomolecule charge on the surface is compensated by the charge in the FETand in the double layer.

Figure 3.2: Figure showing the surface charge densities used to model the system:the charge induced in the nanowire, �FET, the charge of the adsorbed biomolecules,�S, and the charge of the double layer of ions, �DL, are modelled as surface charges.Source [2], �gure edited.

The above relation can be derived by enclosing the entire system with aGaussian surface. The electric �elds are screened by the ions in the solution.If one chooses the enclosing Gaussian surface such that one end is in theregion where all the �elds are screened and the other end in the metal, thereare no �elds penetrating the surface. Gauss theorem then implies that thetotal charge inside the surface is zero.

Assuming there are no other charges present in the system except the ionsin the electrolyte and the target molecule charge, thereby assuming that theoxide is not charged, and assuming that the metal FET and the electrolytesolution are grounded, V (a) = V (1) = 0, the charge per unit length beforebiomolecule attachment was �FET = 0. After attachment the charge in theFET is �FET. Therefore the charge induced in the FET upon biomoleculeattachment is ��FET = �FET. For simplicity the assumption will be madethat the FET and the electrolyte solution are indeed grounded. In the �nalsection it will be discussed what the e�ect is of a potential di�erence betweenthe FET and the electrolyte.

In the following sections the Surface Charge Model will be solved forthe at, section 3.1, and the cylindrical FET, section 3.2. The cylindricalFET results, which are approximate results, will be compared to the more

THE SURFACE CHARGE MODEL 37

accurate results obtained by �nite element simulations.

3.1 The Surface Charge Model for the at sensor

In this section an expression will be derived for the charge induced in the at FET sensor, for a given surface charge. The system is shown in �gure3.3.

Figure 3.3: The at FET biosensor. The biomolecule surface charge is indicatedwith the blue layer, the double layer is formed by the ions, and the green layerrepresents the induced charge in the FET.

The results of the previous chapter can be directly applied to this system.Recall that the width of the at device was chosen to be 2�b, to be able tocompare the results to the cylindrical FET. With the use of section 2.4, andrealizing that the potential across the device is given by b �a = b (see�gure 3.3), the expression of the surface charge per unit length of the deviceinduced in the at FET is found to be

�FET =2�bC atkBT

eb: (3.2)

The charge in the double layer was derived in section 2.5. For an interfacepotential equal to b, the charge per unit length of sensor in the double layeris

� atDL =�4�b�0�r;wkBT

e� sinh

�b

2

�: (3.3)

This can be rewritten to obtain

�FET = � �

� atox

b; (3.4)

and

� atDL = �2�� atb sinh

�b

2

�; (3.5)


where the following de�nitions have been used

� � 2��0kBT

e; (3.6)

� atox � ��0bC at

=toxb�r;ox

; (3.7)

and�� atb � b�r;w�: (3.8)

Equation (3.7) was obtained by using the explicit expression of the capac-itance of the at metallic FET (equation (2.10)). The above expressionswere chosen in such a way that � can be considered to be a constant, sinceT is considered to be constant. As discussed before, the variation of T islimited due to the sensitivity of biosystems to temperature. The form of � atox

is chosen such that it gives the relative thickness of the oxide layer. Theoxide thickness is made dimensionless by dividing it by b. In electrostaticsthe relevant length scale is related to the dielectric constant, since the �eldsin the system are scaled by the dielectric constant. Therefor the expression� atox is inversely proportional to the dielectric constant, �r;ox. The same lineof reasoning holds for �� atb . The inverse of � is the Debye length. This isscaled with another length scale in the system, b, and the dielectric constantof the medium, �r;w.

The above expressions can be substituted into equation (3.1), to obtain:

� �

� atox

b + �S � 2�� atb sinh

�b

2

�= 0: (3.9)

This equation can be solved numerically for b. This can then be used withthe capacitance to �nd the charge induced in the FET, which is the quantityof interest. The numerical solution is plotted as a function of the biomoleculecharge per unit length of the sensor for two di�erent ion concentrations in�gure 3.4. This �gure shows that for increasing surface charge the potentialat the surface of the sensor increases. This is an important result sinceit describes the sensing mechanism. Increasing the surface potential willincrease the charge induced in the FET, which changes the conductivity ofthe device. Attachment of charged biomolecules will increase this surfacecharge per unit length on the sensor, and thereby the conductivity of thedevice. The second interesting aspect shown in this �gure is that for a largerion concentration the potential at the surface of the sensor increases. Thisindicates that due to screening by the ions in the solution the sensitivity ofthe sensot is less. Lastly the �gure shows that for the large ion concentrationsin which we are most interested, the potential stays below 1.


Figure 3.4: The dimensionless potential as a function of the surface charge fortwo di�erent ion concentrations. The ion concentration of interest is 0:1 M. Theblack dots indicate what value of the sensor surface charge is realistic in biosensingexperiments (see the table at the end of chapter 2 for the values of the variables).

The next two subsections will be about making approximations whichenable us to solve for b analytically. The two approximations are b � 1and b � 1. Although it was seen in �gure 3.4 that the approximationb � 1 is a poor approximation for biomolecule sensing, the results for thisapproximation will be given in section 3.1.2 for completeness, and a betterunderstanding of the system. In the end of these subsections the resultsof the two approximations are plotted as a function of the surface charge,together with the numerical solution (�gure 3.7).

3.1.1 Small potentials

Assuming that the surface potential is small, b � 1, equation (3.9) can beapproximated by

� �

� atox

b + �S � �� atb b = 0: (3.10)

This equation can easily be solved for b, yielding

b =�S

�� atb + �=� atox

: (3.11)

Substituting this result back into equation (3.4), and multiplying by thewidth of the device, we obtain the charge per unit length induced in theFET

� atFET = � �

� atox

b

= � 1

1 + �� atb � atox

�S: (3.12)


This equation shows that the charge induced in the FET is linearly depen-dent on the surface charge. The slope of this proportionality depends onthe physical variables via �� atb and � atox . If the slope is large, a small in-crease in the surface charge induces a large amount of charge in the FET,and the sensitivity of the biosensor is high. This suggests the device willperform optimal for a small oxide thickness, a large screening length in theelectrolyte, small dielectric constant of the electrolyte solution, and a largedielectric constant of the covering oxide.

For �� atb � � atox equation (3.12) becomes �FET = ��S. The charge onthe surface is then completely compensated by the charge induced in thenanowire, and the e�ect of screening is negligible. This is the upper limitof the charge induced in the FET. In this region the charge induced in thenanowire is independent of the inverse Debye screening length, �� atb , and therelative oxide layer thickness. This dependence is shown in the left of �gure3.5 (a.).

For �� atb � � atox equation (3.12) becomes ��S=�� atb � atox , where the de-nominator is large. The charge induced on the FET vanishes, indicatingthat the surface charge is (almost) completely compensated by the doublelayer. The dependence on the inverse Debye screening length and the rela-tive oxide thickness is now simple, namely it is proportional to the inverseof these variables, as seen in the right of �gure 3.5 (a.).

A large value of �� atb can be achieved by increasing the ion concentrationof the electrolyte. This decreases the induced charge in the FET, and reducesthe sensitivity of the sensor. It is therefore often said that screening by theions in the electrolyte solution is a large problem for biosensing, and that theion concentration should be kept as low as possible. In biological systemshowever, this is not always possible. The value of �� atb for the biosensingsystem of interest is large, and therefore the approximation �� atb � � atox isthe appropriate one here. Figure 3.5 (b.) shows this dependence of theinduced charge in the FET on �� atb . The dependence on � atox is the same, aswas seen in equation (3.12), and will therefore not be shown here.Using equations (3.7), (3.8), and (3.12), the explicit dependence of the in-duced charge in the metallic FET on physical variables is obtained

� atFET = � 1

1 + �tox(�r;w=�r;ox)�S: (3.13)

The dependence on the inverse Debye length �, can be easily understoodby realizing that screening plays a larger role when the ion concentration islarger.

To understand the dependence on tox one should realize that the poten-tial drop across the FET oxide and in the electrolyte double layer is thesame. The two regions have the same boundary conditions, zero potentialon one side, and b on the other side, at the oxide-electrolyte interface. Nowimagine a situation in which the amount of charge induced in the FET is


a.)

b.)

Figure 3.5: (a.) The charge induced in the at FET, �FET, as a function of �� atb ,the inverse of the relative Debye length, for b � 1. (b.) A realistic value of �� atb

for biosensing is indicated by the black dots. To span the complete range of �� atb

the ion concentration was varied from 0.001 M to 0.4 M. All other variables werestandard as given in the table at the end of chapter 2.

the same as the amount of charge in the double layer. If we now make theoxide layer thinner, the electric �eld in the oxide layer has to be larger, inorder to make the same potential drop possible across this reduced distance.The electric �eld can be made larger by increasing the amount of charge inthe FET. Thus a reduced oxide thickness tox increases the induced charge�FET.

Via Poisson's equation it is seen that the value of the electric �eld due toa certain amount of charge depends on the dielectric constant. The depen-dence on the dielectric constants in equation (3.13) is therefore a necessaryconsequence.

3.1.2 Large potentials

In �gure 3.4 we saw that for decreasing ion concentrations or increasingvalues of the surface charge, the potential at the sensor surface increases. Inthis subsection the induced charge will be derived assuming a large potential.


In the large potential approximation, b � 1, equation (3.9) becomes

�S � �� atb exp

�b

2

�= 0; (3.14)

where the �rst term in equation (3.9) was left out, since exp(b) � b forlarge b, and it was assumed that �� atb 6= 0. Note that b was assumed tobe positive, for a large negative potential some signs will be di�erent. If wesolve the above equation for b, we get

b = 2 ln

��S

�� atb

�; (3.15)

Substituting this into equation (3.4), and multiplying with the width of thedevice, we obtain the charge induced on the at FET per unit length as afunction of �S,

� atFET = � 2�

� atox

ln

��S

�� atb

�

= �4�b�0�r;ox kBTetox

ln

��S e

2�b�0�r;wkBT�

�: (3.16)

It is seen that for larger �� atb , so for a larger ion concentration, the chargeinduced in the FET is smaller, showing again that screening decreases thesensitivity. The dependence on �� atb and � atox is shown in �gure 3.6 (a.) and(b.) respectively. The value of the surface charge was taken to be 100 timeslarger than one would expect for biosensing, otherwise the approximationthat was used, b � 1, would not hold. It is seen that also in this limitscreening has a negative e�ect on sensing, and that it is bene�cial to havea small value of � atox , which corresponds to having a device with a largecapacitance.

3.1.3 Comparing the small and the large potential regime

In the previous sections is was shown that for small potentials the depen-dence of the charge induced in the FET, �FET, on the surface charge, �S,is linear (equation (3.12)), while for large potential this dependence is log-arithmic (equation 3.16). This can be explained by the fact that the e�ectof screening is exponential in the large potential limit, and therefore hasmore in uence in this limit. Sensing is thus a�ected more by screening inthe large potential limit than in the small potential limit. A small change inthe linear regime causes a much larger change in the charge induced in theFET, than in the logarithmic regime. This means that the sensor is moresensitive in the small potential regime.

The charge induced in the FET both in the approximation b � 1 andin the approximation b � 1 is plotted in �gure 3.7 as a function of the


a.)

b.)

Figure 3.6: The charge induced in the at FET, �FET, as a function of (a.) �� atb

and (b.) � atox , in the regime where b � 1. The surface charge is taken to be 100times larger than realistic, to make sure the approximation is valid. The values ofthe other variables are chosen to be standard (see table at the end of chapter 2).

surface charge. The green line is the result which was found numerically.The black dots indicate a realistic value of the surface charge. As can be seen,this value is well within the regime where the small potential approximationholds, and the sensor is expected to operate in its sensitive regime.

The crossover point between the region where the small and where thelarge surface potential result is more applicable can be found by consideringthe ratio of the two results

� at;largeFET

� at;smallFET

=2(1 + �� atb � atox )

�� atb � atox

lnh

�S�� at

b

i�S

�� atb

; (3.17)

taking the derivative with respect to �S, and setting this equal to zero. Thisgives an estimated crossover point at �S = �� atb e, with e the base of thenatural logarithm.

The dependence on the two variables �� atb and � atox in the regime b � 1is di�erent from that in the regime b � 1. In both cases however it is


Figure 3.7: The numerical result of the charge induced in the at FET, �FET,as a function of the charge on its surface, �S, and the analytical results using theapproximations b � 1 and b � 1, for two values of the ion concentration. Left:0.001 M. Right: 0.1 M, which is more realistic for biosensing. All other variablesare taken to be standard (see the table at the end of chapter 2). The realistic valueof the biomolecule surface charge, �S, is indicated by the black dots.

bene�cial to have a small value of �� atb , and a small value of � atox , implyinga small ion concentration and a thin oxide layer.

3.2 The Surface Charge Model for the cylindrical

sensor

In this section the charge induced in the cylindrical FET sensor will becalculated in the Surface Charge Model. The system is shown in �gure 3.8.

Figure 3.8: The nanowire sensor in the SCM model. The biomolecule charge isshown as a blue layer on the sensor surface. The green layer represents the chargeinduced in the FET. The ions in the double layer charge are shown with red andblue dots, representing the negative and the positive ions respectively.

The charge induced in the cylindrical FET as a function of the surfacepotential was given in the previous chapter (equations (2.9) and (2.11)).


With the FET surface potential b, the charge in the FET is

�FET =CcylkBT

eb: (3.18)

The charge in the double layer is given by equation (2.26), and gives

�cylDL =�2�0�r;wkBT

e� sinh

�b

2

�241 + ( (�b))�2 � 1

cosh2�b

4

�35

12

: (3.19)

To calculate the charge per unit length of FET device, the surface chargedensities have to be multiplied by di�erent factors, one factor for the chargeinduced in the FET, �FET, and a di�erent factor for the charge in the doublelayer, �cylDL. This is due to the fact that the area where the charge inducedin the FET is located is smaller than the area of the electrolyte interface.Multiplying equation (3.18) by 2�a, equation (3.19) by 2�b, and rewritingthese expressions we obtain

�FET = � �

� cylox

b; (3.20)

and

�cylDL = �2�� atb sinh

�b

2

�241 + ( (�b))�2 � 1

cosh2�b

4

�35

12

; (3.21)

with � and �� atb de�ned in equations (3.6) and (3.8), and

� cylox � ��0aCcyl

=ln(1 + tox

a )

�r;ox: (3.22)

Substituting this into equation (3.1), we obtain

� �

� cylox

b + �S � 2�� atb sinh

�b

2

�241 + ( (�b))�2 � 1

cosh2�b

4

�35

12

= 0; (3.23)

This equation can be solved numerically for b. The result is given in�gure 3.9. It can be seen that also for the cylindrical sensor the potentialremains small for the relevant value of the ion concentration and the surfacecharge. It is seen that for lower ion concentrations, the large potentialregime, b � 1, is a valid approximation.

The numerical solution for b can be used to �nd the induced chargein the FET as a function of the surface charge. In the following sections


Figure 3.9: The dimensionless potential as a function of the surface charge fordi�erent ion concentrations, in the cylindrical geometry. All other variables weretaken to be standard (see table at the end of chapter 2)

approximations will be used, such that an analytical solution of the aboveequation is possible. The two approximations are b � 1 and b � 1. Likefor the at FET in the previous section, both the numerical and the twoapproximate results are plotted at the end of this section in �gure 3.10.


Under the assumption b � 1 equation (3.23) can be approximated by

� �

� cylox

b + �S �� atb b

(�b)= 0: (3.24)

The potential b can then be expressed as

b =�S

�=� cylox + �� atb = (�b): (3.25)

Substituting this in equation (3.18) and multiplying this by the circumfer-ence, 2�a, we �nd the charge per unit length induced in the cylindrical FETto be

�cylFET = � 1

1 + � cylox �� atb = (�b)�S

= � 1

1 + ��cylb � cylox

�S; (3.26)

where

��cylb =�� atb

(�b): (3.27)


This result is similar to the result for the at FET (equation (3.12)).

The di�erence is that � atox and �� atb are replaced by � cylox and ��cylb .Using equations (3.22), (3.27) and (3.8), the charge induced per unit

length in the cylindrical FET can be written explicitly as

�cylFET = � 1

1 + (�r;w=�r;ox) (�b= (�b)) ln�1 + tox

a

��S: (3.28)

This simple expression shows that in order to get a large induced charge inthe cylindrical FET, just as for the at FET, it is bene�cial to create a devicein which the dielectric constant of the surrounding medium is comparableto that of the oxide, the Debye screening length is large compared to thedevice radius, and the oxide thickness is small compared to the inner FETradius.


Under the assumption that b � 1 equation 3.23 becomes

� �

� cylox

b + �S � �� atb exp

�b

2

�241 + ( (�b))�2 � 1

exp�b

2

�=4

35

12

= 0: (3.29)

Assuming (( (�b))�2 � 1) � exp (b=2) =4, which is a realistic assumptionwhen b � 1, the part between the square brackets will be approximatelyequal to one. Also for large potentials b is much smaller than exp (b) andthe �rst term drops out, leading to [2]:

�S � �� atb exp

�b

2

�= 0: (3.30)

The solution of this equation gives the expression for b, as also found byNair an Alam [2]:

b = 2 ln

��S

�� atb

�; (3.31)

Substituting this in the expression for the charge induced in the FET, equa-tion (2.8), we obtain the result for the charge induced on the cylindricalFET per unit length

�cylFET = � 2�

� cylox

ln

��S

�� atb

�

= �4��0�r;ox kBTln(1 + tox

a )eln

��S e

2�b�0�r;wkBT�

�: (3.32)

This is again similar to the result of the at FET. The dependence on �� atb is

the same, and the role of � cylox is the same as that of � atox in equation (3.26).



The numerical solution, and the approximate solutions are shown in �gure3.10 for two di�erent values of the ion concentration. It is seen that, justas for the at FET, the relevant regime is the small potential regime. Forsmaller values of the ion concentration, the large potential approximationbecomes more important. The result in equation (3.26) showed that theinduced charge in the FET has a linear dependence on the surface chargein the small potential regime, and equation (3.32) showed a logarithmicdependence in the large potential limit. Therefore the small potential regimeis also the sensitive regime for the cylindrical FET.

Figure 3.10: The numerical results of the charge induced in the cylindrical FET,�FET, as a function of the charge on its surface, for two di�erent values of the ionconcentration. The value 0.1 M is realistic for biosensing. The analytical results forthe approximations b � 1 and b � 1 are also shown. The values of the othervariables are the standard values (see table at the end of chapter 2).

3.3 Comparing the e�ect of surface charge for the

at and the cylindrical sensor

In the previous section it was seen that for both the small potential approx-imation as the large potential approximation the expression of the inducedcharge in the FET per unit length has the same form in the cylindrical sys-tem as it has in the at system. The only di�erences are due to the factthat the variables � cylox and � atox are not the same, and that ��cylb and �� atb arenot the same. These di�erences are caused by the di�erences in capacitanceand the di�erence in the double layer of the two geometries respectively. Inthis section these di�erences will be studied.

Comparing equation (3.27) to equation (3.8), we �nd that the di�erence

between ��cylb and �� atb is only a factor ( (�b))�1. As was seen in chapter 2,

�gure 2.18, ( (�b))�1 never reaches values below one. We �nd ��cylb =�� atb =


( (�b))�1 � 1, which makes the charge induced in the cylindrical FET,

�cylFET, smaller than the charge induced in the at FET, � atFET.

Comparing equation (3.22) to equation (3.7), we �nd that the di�erence

between � cylox and � atox is due to the di�erence in the capacitance per unitlength of the devices. ~C at = 2�bC at and ~Ccyl = 2�aCcyl. In �gure 2.13it was seen that the capacitance per unit length of the cylindrical FET issmaller than that of the at FET. We �nd � cylox =� atox = ~C at= ~Ccyl � 1, which

also makes �cylFET smaller than � atFET.

So in general a certain amount of surface charge will induce less chargein the cylindrical FET than in the at FET. This means that the cylindricalFET is less sensitive to the attachment of a uniform surface charge.

a.) b.)

c.) d.)

Figure 3.11: The charge induced in a at (green line) and a cylindrical (red line)FET due to adsorption of the same amount of surface charge as a function of (a.)tox=a, and (b.)�b. In �gure (c.) and (d.) the ratio of these two results is shown.

The di�erence between the results of the two geometries for the case thatb � 1 is shown in �gure 3.11. The �rst �gure shows the charge inducedin the at (green line) and the cylindrical (red line) FET as a function oftox=a, which represents the di�erence in capacitance. A realistic value oftox=a is 1/2. The charge induced in the at FET is seen to be only 77% ofthat induced in the cylindrical FET. The plot as a function of �b shows thedi�erence due to the di�erent double layers in the two geometries. It shouldbe noted that for small values of �b the analytical result of the cylindricalgeometry is not accurate due to the fact that the assumption was used inthe derivation that this was large [13]. Therefore the left of the plot is notaccurate. For biosensing we are usually not interested in this regime, andno further investigation is necessary.

It was shown in section 2.5 that for the relevant value of �b there wasonly a small di�erence in the double layer charge, and in fact the di�erence


between the two geometries is therefore mainly caused by the di�erencein capacitances. This can be seen in �gure 3.12, where in �gure (a.) theexpression of the double layer in the cylindrical geometry was set equal tothat in the at geometry by imposing = 1. A realistic value of tox=a is1=2. It can be seen in �gure 3.12 that for this value 20 % less charge isinduced in the cylindrical FET than in the at FET, even without takinginto account the e�ect of the di�erence in double layer.

In �gure 3.12 (b.) the capacitance of the cylindrical FET was assumedto have the value of the capacitance of the at FET. A realistic value of�b is 15.5. For this value the di�erence between the charge induced inthe at FET and the charge induced in the cylindrical FET is seen to beonly 6 percent. This was without including the e�ect of the di�erence incapacitance.

a.) b.)

Figure 3.12: The ratio of the charge induced in the cylindrical FET and thecharge induced in the at FET. (a.) This ratio as a function of tox=a while theexpression for the surface charge and the double layer charge is set the same inboth geometries. (b.) This ratio as a function of �b while the capacitances of the at and the cylindrical devices are taken to be the same.

Considering the case that b � 1, it is seen that the only di�erencebetween the results of the two geometries is in the capacitances of the twodi�erent FETs. The ratio of the charge induced in the cylindrical sensorand the charge induced in the at sensor exactly follows the ratio of thecapacitances given in �gure 2.13 (b.), �cylFET=�

atFET = ~Ccyl= ~C at. In this

regime the di�erence in the charge in the double layer of the two geometrieshas no in uence.

It can be concluded that electrostatically, the cylindrical FET sensor isless sensitive to a uniform biomolecule surface charge than the at FETsensor. This is due to both the di�erence in capacitance and the fact thatscreening of the double layer in the cylindrical geometry has a strongerin uence than in the at geometry. However, for a nanowire with realisticdimensions in a physiological solution, the di�erence in double layer chargeis only small. In this realistic biosensing regime the di�erence therefore ismainly caused by the di�erent capacitances.


3.4 Applying a voltage to the sensor

In the previous sections it was always assumed that the FET was grounded.When the FET is not grounded the results change, since the charge inducedin the FET is no longer only dependent on the surface potential, but alsoon the FET potential. The expression for the induced charge is then for the at FET

�FET = 2�bC(b �a)kBT

e: (3.33)

Since the capacitance of the FET now has to be multiplied with the potentialdi�erence across the FET oxide (equation (2.8)), the equation for the atFET becomes (equation (3.9))

�

� atox

(b �a) + �S � 2�� atb sinh

�b

2

�= 0; (3.34)

The above equation can be rearranged to �nd the following expression

�

� atox

b + ~�S � 2�� atb sinh

�b

2

�= 0; (3.35)

with

~�S = �S +�

� atox

a: (3.36)

Equation (3.35) is similar to equation (3.9), with the di�erence that �Sis replaced by ~�S. The e�ect of the FET potential is seen to be equivalentto the e�ect of a surface charge. In �gure 3.13 it is shown that for a FETvoltage of 0.7 V the e�ect of the potential is just as large as the e�ect of thesurface charge.

Figure 3.13: The e�ect of a FET potential is equivalent to that of the surfacecharge. The value of �a=�

atox is equal to the value of the surface charge for a FET

potential of 0.7 V.


Replacing �S by ~�S leaves all the calculations the same. We can thereforesimply take all the previously derived results, and use equation (3.36) toobtain the dependence on a.

In the small potential limit, b � 1, we obtain (equation (3.12))

� atFET = � 1


(�S +�

� atox

a): (3.37)

It is seen that the charge induced in the FET is still linearly dependent onthe surface charge when b � 1. However, the FET potential a determinesin which regime the FET is operating, and it can no longer be assumed thatthe biosensing system is in the small potential regime. The FET potentialcan be used to shift between the small and the large potential regime.

In the large potential approximation, b � 1, we obtain (equation(3.16))

� atFET = � 2�

� atox

ln

"(�S +

�� atox

a)

�� atb

#: (3.38)

In �gure 3.14 this result is illustrated. Depending on the value of the FETpotential, a, a certain change in surface charge, �S, will induce a di�erentamount of charge in the FET. The FET potential can thus be used to tunethe sensor to be more sensitive.

In the article written by Nair and Alam [2] the large potential approx-imation is calculated. It was shown in this chapter that for reasonablebiomolecule densities and charge the value of the surface charge density issuch that the biosensor operates in the small potential regime. For largersurface charge densities or if the FET potential is not equal to zero, the op-eration of the biosensor moves towards the large potential regime. The largepotential regime is, however, not the most sensitive regime of operation ofthe biosensor, and it is recommended that the sensor operates in its smallpotential regime.

The above results (equations (3.37) and (3.38)) are also valid for the

cylindrical case, but then � atox and �� atb are replaced with � cylox and ��cylb .


Figure 3.14: The charge induced in the FET upon attachment of a surface charge,�S, is dependent on the FET potential, a. The position on the x-axis is determinedby the FET potential a. The interval on the x axis is given by �S. It is seen thatfor larger values of the FET potential less charge is induced in the FET uponattachment of surface charge.


Chapter 4

The Excluded Volume Model

The �eld-e�ect transistor (FET) sensor detects changes of the potentialon its surface. In the previous chapter it was shown how attachment ofcharged molecules a�ects the FET surface potential. In this chapter it willbe shown how another property of the biomolecules, their volume, in uencesthis potential [3].

When biomolecules attach to the sensor surface, they push away a certainamount of electrolyte. This e�ect is illustrated in �gure 4.1. The presenceof the biomolecules excludes the presence of the electrolyte, therefore thismodel is called the Excluded Volume Model (EVM).

By including the volume of the biomolecules in the model, two e�ectshave to be taken into account. The �rst one is a change in the dielectricconstant. Since the dielectric constant of the biomolecule is di�erent fromthat of the electrolyte, attachment of biomolecules changes the dielectricconstant in a thin layer around the FET.

A second e�ect is a decrease of the amount of ions close to the sensorsurface. Ions in the electrolyte in the region close to the sensor surface arepushed away by the biomolecules. If a double layer is formed in a thin layerclose to the sensor surface (for example due to a �xed surface charge, adi�erent FET potential, or a back gate), the removal of these ions will causea rearrangement of the charges in the vicinity of the sensor.

These two e�ects are independent of the charge of the biomolecule, sohypothetically in this way even the presence of neutral biomolecules can bedetected. The question remains whether the in uence of the volume of thebiomolecules is strong enough to detect the attachment of biomolecules?Is this a strong e�ect or can it be neglected compared to the e�ect ofbiomolecule charge?

In this and the following chapter the e�ects of the biomolecule volumewill be studied, and the results will be compared to the results of the Sur-face Charge Model (chapter 3). This chapter will be dealing with a densebiomolecule layer in which all the volume in a thin layer on the sensor sur-

55

56 THE EXCLUDED VOLUME MODEL

face is occupied by biomolecules. This will provide an upper limit of thee�ect of biomolecule volume. In this chapter most results can still be de-rived analytically. In the next chapter a partial occupation of this layer isstudied using numerical calculations.

Figure 4.1: The e�ect of the volume that is occupied by biomolecules. When themolecules attach to the sensor surface, they push away the electrolyte in a thinlayer around the sensor, thereby changing the local dielectric constant and the ionconcentration.

The results of the model can be applied to two situations. In the �rst casethe charge on the surface charge is due to attachment of biomolecules. Theresults will answer the question how big the error is when the biomoleculesare modelled as point charges, instead of molecules with a certain size. In�gure 4.2 a biomolecule is shown and how it is modelled when both thee�ect of charge and the e�ect of volume are taken into account.

Figure 4.2: A biomolecule modelled as a big volume with a point charge close tothe sensor surface.

In the second case the biomolecule is modelled as a volume withouta charge, while there are some charges present in the system on the sensorsurface that are �xed (see �gure 4.3). If a device is made with a �xed surfacecharge, biomolecules can be detected even without taking into account thee�ect of the biomolecule charge. We want to answer the question whetherthis e�ect is more pronounced than the charge e�ect in certain cases.

The charge induced in the FET is related to the other charges in the

THE EXCLUDED VOLUME MODEL 57

Figure 4.3: A biomolecule modelled as a big volume without charge, and a sepa-rate surface charge �xed to the sensor surface.

system by an equation that is similar to equation (3.1), with the di�erencethat �S is no longer necessarily attributed to the charge of the attachedbiomolecules, but it can also be a �xed surface charge (e.g. oxide or capturemolecule charge)

�FET + �S + �DL = 0: (4.1)

This equation can be used to �nd the charge induced in the FET, �FET.Changing any of the three above surface charges in uences the whole system,thereby changing the charge induced in the FET. Recall that in the previouschapter the attachment of biomolecules changed the surface charge, �S, whileleaving the expression for the charge in the double layer the same. Includingthe volume of the biomolecules will have an in uence on the charge in thedouble layer, �DL.

4.1 The Excluded Volume Model for the at sen-

sor

In this section an analytical expression is derived for the charge induced inthe at FET when a surface charge and a dense biomolecule layer are presenton the surface of the FET (see �gure 4.4). In the dense biomolecule layerno ions are present, and the dielectric constant is �r;bio. The thickness of thelayer will be represented by tBIO = c � b, and again the FET is grounded,Va = 0.

In chapter 2 expressions were derived for the charge in the FET and thecharge in the double layer. In �gure 4.4 it can be seen that now the surfacepotential that determines the charge induced in the FET is b, and thatthe potential of the electrolyte interface, that determines the charge in thedouble layer is c. Equation (4.1) then becomes

�FET(b) + �DL(c) + �S = 0; (4.2)

which can not be solved without an extra relation between b and c. Thisrelation is found by the following calculation.


Figure 4.4:

The electric �eld inside the dense biomolecule layer is

E =�FET(Vb) + �S

�0�r;bio: (4.3)

Integration over x from b to c yields

Vc � Vb = �� atFET + �S�0�r;bio

tbio; (4.4)

or

c �b = �� atFET + �S2�b�0�r;bio

�tbioe

kBT

�: (4.5)

Inserting into this equation the relation between the charge induced in theFET and the FET surface potential (equation (2.9)),

b =e

kBT

� atFET

2�bC at(4.6)

and solving for �FET, an expression for �FET is obtained that is dependenton c instead of on b

� atFET = ��S +kBTe c 2�b �0�r;bio=tbio

1 +�0�r;bio=tbio�C at

= ��S +

�� atbio

c

1 + � atox =� atbio

; (4.7)

with � and � atox de�ned in equation (3.6) and (3.7), and

� atbio �tbio

b �r;bio: (4.8)


As mentioned before, the double layer charge depends on the interfacepotential which is c. The charge in the double layer is

� atDL =�4�b�0�r;wkBT

e� sinh

�c

2

�

= �2�� atb sinh

�c

2

�; (4.9)

with �� atb de�ned in equation (3.8).Substituting equations (4.7) and (4.9) into equation (4.1) we obtain

��S +

�� atbio

c


+ �S � 2�� atb sinh

�c

2

�= 0: (4.10)

In �gure 4.5 the numerical solution of this equation is given as a functionof the surface charge for three di�erent values of the biolayer thickness. Itis seen that the value of the dimensionless potential, c, is smaller than 1for any surface charge value that is realistic for a biomolecule charge. Thevalues of the variables were also chosen to have values that are realisticfor biosensing experiments. If other charges are present in the system thevalue of the surface charge may be much larger. For increasing surfacecharge the potential increases rapidly, and the potential will reach valueslarger than one. For larger biomolecule layer thicknesses the large potentialregime will be reached less rapidly. Note that in the �gure the potential atposition c is plotted, while in the previous chapter the potential at positionb was considered. For increasing biomolecule layer thickness, the potentialat position c decreases, while the potential at position b increases.

Figure 4.5: The numerical solution of equation (4.10) is given as a function ofthe surface charge for di�erent values of the biolayer thickness. It is seen that c

stays well below 1. The values of the other variables are chosen such that they arerealistic for biosensing applications (see standard values in the table at the end ofchapter 2).

In the following sections equation (4.10) will be solved analytically forc using the approximations c � 1 and c � 1 respectively. As discussed


above, c will be small in the case of a surface charge due to biomoleculeattachment and under these circumstances the approximation c � 1 willbe a good one. The other approximation, c � 1, will be given since itmay be bene�cial to design the sensor with a much larger surface charge forvolume detection, as will become clear later on, and this gives rise to largepotentials. Besides that, a FET that is not grounded can also give rise tolarge potentials.


Applying the approximation c � 1 to equation (4.10), we get

��S +

�� atbio

c


+ �S � �� atb c = 0; (4.11)

for which the solution of c is

c =� atox =�

1 + �� atb (� atbio + � atox )�S: (4.12)

Substituting this back into the expression for �FET (equation (4.7)) thefollowing expression is obtained

� atFET = � �S


1 +

� atox =� atbio

1 + �� atb

�� atbio + � atox

�!: (4.13)

This can be rewritten as

� atFET = � 1


�1 +

�� atb � atbio

1 + 1=(�� atb � atox ) + � atbio =� atox

��S: (4.14)

It is now easily seen that in the limit � atbio = 0 the result of the calculationsof the Surface Charge Model are obtained (equation (3.12)). Just as inthe Surface Charge Model, again � atFET is linearly dependent on the surfacecharge, �S, and decreases for increasing �� atb . This means that less charge isinduced in the FET when screening is enhanced. In the case of the excludedvolume model this e�ect however is less strong.The ions are excluded fromthe region close to the surface, reducing the e�ect of screening. This isexplained below.

The second term in brackets in equation (4.13) is much smaller than1. This can be seen since the value of �� atb is large (�� atb � 1200, equation(3.8)) compared to the values of � atbio (� 1=15, equation (4.8)) and � atox (�1=12, equation (3.7)), while � atbio and � atox have the same order of magnitude.Therefore � atox =� atbio is much smaller than �� atb


�. The second


term in brackets is much smaller than 1 and can be neglected compared tothe �rst term. Equation (4.13) therefore becomes

� atFET � ��S


; (4.15)

for the values of the physical parameters relevant for biosensing. Note thatif � atbio � � atox , the above relationship no longer holds. The dependence on�� atb was located in the second term in brackets of equation (4.13). Thisterm was shown to be negligible, and therefore it can be concluded that thein uence that �� atb has on � atFET is very small. The in uence of screening isshown to be reduced.

The dependence on the biomolecule layer thickness, and the oxide thick-ness is also clear in the above equation. If the biomolecules layer thicknessis increased the charge in the FET increases. For increasing oxide thicknessthe charge in the FET decreases.

Now the two applications of the Excluded Volume Model result willbe studied. In the �rst case (EVM1), where all the surface charge is at-tributed to the biomolecules, there was no charge on the sensor surfacebefore biomolecule attachment. The induced charge upon attachment ofthe biomolecules is

��EVM1; atFET = �EVM; at

FET

= � �S


1 +

� atox =� atbio

1 + �� atb


�!

� � �S


: (4.16)

The dependence of the exact result above on the relative biomolecule layerthickness, � atbio , is shown in �gure 4.6 (a.). It is seen that for increas-ing biomolecule layer thickness more charge is induced in the FET uponbiomolecule attachment. The dependence on �� atb is shown in �gure 4.6 (c.).It is seen that, as noted before, when the ion concentration in the electrolyteis larger, less charge is induced in the FET. For increasing biomolecule layerthickness the in uence of the ion concentration is less and less. For realisticvalues of the ion concentration the value of �� atb is 1250 (equation 3.8), andit is seen that in this part of the graph the lines are almost horizontal, andsmall uctuations in the ion concentration are not noticeable compared tothe e�ect of the excluded volume. Figure 4.6 (e.) shows that for increasingoxide thickness less charge is induced in the FET. A realistic value of � atox is0.08. This all suggests that just like for the SCM, for the EVM the sensorwill respond best if the ion concentration is small and the oxide layer is thin.

Figures 4.6 (b.), (d.), and (e.) show the ratio of the charge induced uponbiomolecule attachment if the biomolecules are modeled as a surface charge


and a volume layer, and if the biomolecules are only modeled as a surfacecharge. The former is the result of the Excluded Volume Model. The latteris the Excluded Volume Model result when � atbio = 0, and is equivalent tothe Surface Charge Model result. Using equation 4.14 the correspondingexpression is found to be:

��EVMFET

��SCMFET

= 1 +�� atb � atbio


(4.17)

In this equation the second term is large compared to the �rst. Realizingthat � atbio =�

atox � 1, and that �� atb � atbio and �� atb � atox are large and have a

value of about 100, it can be concluded that the second term has a valueof about 50. This means that if the attached biomolecules are modelled asonly a surface charge, while in fact they do not only have a charge but alsoform a dense surface layer on the sensor surface, the result that is found isabout a factor 50 too small. The error in modelling the biomolecule as apoint charge without including any volume e�ect could therefore lead to aserious underestimation of the charge induced in the FET, and therefore ofthe sensor signal.

In �gure 4.6 (b.) equation (4.17) is plotted as a function of the layerthickness. It is seen that for a layer thickness of 10 nm (correspondingto � atbio = 0:33), the error is a factor 84, for a layer thickness of 2 nm(corresponding to � atbio = 0:067) the error is a factor 47, and for a layerthickness of 0.94 nm (corresponding to � atbio = 0:031) the error is a factor 29.This last thickness is chosen such that the excluded volume in the surfacelayer is the same as the total volume of the biomolecules, when a surfacecharge density and a biomolecule volume is chosen as derived in section 2.2(see the table at the end of chapter 2). Or put di�erently, this is the totalvolume of the biomolecules spread out over the sensor surface. In �gure 4.6(d.) it can be seen that the error increases for increasing �� atb , which is forincreasing ion concentration. This could explain (part of) the mysterioussensitivity of nanowire sensors at large ion concentrations [3] [14]. In �gure4.6 (f.) it is seen that the error is larger when the oxide layer is thicker,and this dependence is stronger for larger biolayer thicknesses, meaningthat the negative e�ect of a large oxide thickness is less pronounced when abiomolecule layer is present on the surface.

The results of the second application of the model, in which we inter-pret the result in equation (4.13) as the e�ect of the attachment of neutralbiomolecules on top of a �xed surface charge, will be discussed now. We wantto know how much charge will be induced in the FET upon attachment ofa dense layer of neutral biomolecules. We are interested in the di�erencebetween the result of the SCM and the EVM, since the SCM gives the situ-ation before attachment, and the EVM gives the situation after attachmentof the dense biomolecule layer on the sensor surface. This charge induced in


a.) b.)

c.) d.)

e.) f.)

Figure 4.6: The charge per unit length induced in the at FET, � atFET in theapproximation c � 1 for the �rst application of the model. The result is plottedas a function of (a.) the relative biolayer thickness, � atbio , (c.) as a function of �� atb ,and (e.) as a function of the relative oxide thickness, � atox . The ratio of these resultsand the results of the Surface Charge Model are shown to their right in �gures (b.),(d.), and (e.). These �gures show that the in uence of the biomolecule volume canbe very large, and that for increasing ion concentration the e�ect of an excludedvolume increases. The values of the variables are given in the table at the end ofchapter 2.

the FET upon attachment of neutral biomolecules is

��EVM2FET = �EVMFET � �SCMFET

= � 1


�� atb � atbio


��S;(4.18)

where the surface charge is for example an oxide charge or a capture moleculecharge. The induced charge upon biomolecule attachment is large for asensor with a thin oxide layer, and a thick biomolecule layer, just like in theother models. The dependence on the value of �� atb requires some furtherinvestigation. It was seen that the result of the EVM is almost independent


of the ion concentration. It was also seen that according to the SCM theinduced charge decreases for increasing ion concentration. This suggests thatthe di�erence between the two increases for increasing ion concentrations.

Equation (4.18) can be rewritten to obtain

��EVM2FET = � �S

2=(�� atb � atbio ) + � atox =� atbio + 1=((�� atb )2� atox � atbio ) + 1=(�� atb � atox ) + 1;

(4.19)

which shows that for increasing �� atb the charge induced in the FET uponbiomolecule attachment indeed increases. This time a large value of �� atb

has a positive in uence on the amount of charge induced in the FET uponbiomolecule attachment! This is illustrated in �gure 4.7. It can be explainedby the increased ion concentration, which increases the charge density closeto the sensor surface which is moved away by the biomolecules. This im-plies that the "problem" of a large ion concentration could be used as anadvantage in biosensing. This can have large implications for sensor design.

Figure 4.7: The charge per unit length induced in the at FET upon biomoleculeattachment in the Excluded Volume Model, ��EVM2

FET = �EVMFET � �SCMFET , for c � 1.For an ion concentration of 0.1 M the value of �� atb is 1250.

It was seen in equation (4.18) that the dependence of ��EVM2FET on the

surface charge is linear. This means that when the sensor is charged withmore surface charge, the presence of the volume of the biomolecules is de-tected more strongly. Therefore, one can increase the �xed surface chargeon the surface for a good use of this excluded volume sensing mechanism.

To compare the e�ect of neutral biomolecules with a volume to that ofcharged biomolecules without a volume, the ratio of the above result andthe result of the Surface Charge Model is considered,


�2�EVMFET

�SCMFET

=�EVMFET � �SCMFET

�SCMFET

=�EVMFET

�SCMFET

� 1: (4.20)

This expression is similar to that of �EVMFET =�SCMFET (equation (4.17)). Onlya constant value of 1 has to be substracted, and then all the results areapplicable, and also �gure 4.6 can be used to study the trends. It is seenthat for large ion concentrations the e�ect of a neutral biomolecule layer isabout 50 times larger than the e�ect of a surface charge. This implies thatfor biosensing, the e�ect of the volume of the biomolecules may have a largein uence on sensing.


In the previous section it was shown that increasing the �xed surface charge,can enhance the properties of the biosensor when an excluded volume layer isincluded in the model. A large surface charge increases the potentials, and atsome point the small potential approximation will no longer be valid. In thissection the charge per unit length induced in the at FET for the ExcludedVolume Model will be derived under the assumption that the potentials inthe system are large (c � 1).

Applying the approximation c � 1 to equation (4.10), we get

� �S


+ �S � �� atb exp

�c

2

�= 0; (4.21)

where the second term in equation (4.10) was neglected since exp(c)� c.Solving the above equation for c yields

c = 2 ln

��S

�� atb

�1

1 + � atbio

=� atox

��; (4.22)

which reduces to the result of the Surface Charge Model (equation (3.15)) inthe limit � atbio = 0. Inserting this result in equation (4.7) the charge inducedin the FET per unit length is obtained

� atFET = � �S� atbio

� atbio + � atox

� 2�


ln

��S

�� atb

�1

1 + � atbio

=� atox

��: (4.23)

Just like in the case of small potentials the second term in this equationis much smaller than the �rst term. This is because in order for the largepotential approximation to be valid the surface charge must be large. Sincethe �rst term has a linear dependence on this surface charge and the second


only a logarithmic dependence. Since the second term is negligible comparedto the �rst, the in uence of the ion concentration is small. The aboveexpression can therefore be approximated by

� atFET � ��S�

atbio


; (4.24)

which is the same as the result for the small potential in equation 4.15.The behavior of the charge in the at FET for small potentials and largepotentials is therefore the same as long as the surface charge is large. Notethat without the e�ect of volume, or �bio = 0, the dependence on the surfacecharge was logarithmic in the large potential regime, as also shown by Nair[2]. We have found here that the e�ect of a volume layer gives a linear term.

In the �rst application of the model we have


FET

� � �S� atbio


: (4.25)

It is seen that the result of the large potential approximation is similar tothat of the small potential approximation, and its behavior will not be stud-ied here. The above expression has to be compared to the result withoutthe volume e�ect, the SCM, to obtain the error of not including the e�ectof volume. This SCM expression is di�erent in the large potential approx-imation than in the small potential approximation. The SCM has a resultthat is linear in �S for small potentials and a result that has a logarith-mic dependence on �S for large potentials. This error does therefore nothave the same behavior in the large potential approximation as in the smallapproximation, and must be studied.

��EVM1; atFET

��SCMFET

=�EVMFET

�SCMFET

=� atox

2� ln[�S=(�� atb )]

��S�

atbio


+2�


ln

��S

�� atb

�1

1 + � atbio =� atox

��

� �S� atbio �

atox�


�2� ln[�S=(�� atb )]

: (4.26)

In �gure 4.8 (a.) the dependence of the result on � atbio is shown. Thedependence on the surface charge �S is given in �gure 4.8 (b.). It is seenthat in order for this approximation to be true, the surface charge has to belarge, and the dependence of the ratio on the surface charge is dominated


by the linear term on top. The factor can be varied by changing this valueof the surface charge.

In the second application of the model the charge induced in the FETupon biomolecule attachment is


FET � �SCM; atFET

� � �S� atbio


+2�

� atox

ln

��S

�� atb

�: (4.27)

For an increasing ion concentration the second term becomes smaller. There-fore there is more negative charge induced in the FET upon biomolecule at-tachment if the ion concentration is large. So just like in the small potentialregime, it is also bene�cial to have a large ion concentration in the largepotential regime.

To compare the charge that is induced in the FET by the attachmentof neutral biomolecules to the charge induced by the attachment of chargedbiomolecules modelled as point charges, we need the ratio

��EVM2; atFET

��SCM; atFET

=�EVM; atFET � �SCM; at

FET

�SCM; atFET

=�EVM; atFET

�SCM; atFET

� 1: (4.28)

This is again almost the same as equation 4.26. It is seen that the e�ectof an excluded volume layer can be much larger than the e�ect of a surfacecharge, and can be tuned by varying the surface charge.

a.) b.)

Figure 4.8: The ratio of the charge induced in the at FET, � atFETupon attachmentof a surface charge and a volume layer, and the charge induced in the at FET uponattachment of only a surface charge layer, in the approximation c � 1. (a.) Theresult is plotted as a function of the relative biomolecule layer thickness, � atbio . (b.)The result is plotted as a function of the thickness of the surface charge, �S. Notethat the surface charge is taken to be 100 times larger than before, since this waythe potential is large and the condition c � 1 is satis�ed. The values of the othervariables are as stated in the table at the end of chapter 2.



In �gure 4.9 the numerical result and the result of both approximationsc � 1 and c � 1 are shown as a function of �S. The dependence wasshown to be approximately linear, but it is shown on a log scale here toshow the whole range of possible values of the surface charge and thus thepotential c. It is seen that for realistic values of the physical parametersboth approximations give the same result. Only for very thin biomoleculelayers do the two approximations give di�erent results.

This implies that for the applications of interest, in the EVM we do nothave to distinguish between the regime where c � 1 and c � 1. Forthe results of the SCM, however, the behavior is di�erent in the b � 1and b � 1 regime. Therefor, when comparing the results of the EVM tothe results of the SCM we have to distinguish between two regimes, namelyb � 1 and b � 1.

Figure 4.9: The numerical result of the charge induced in the at FET sensor,�FET (green line), as a function of the charge on its surface, �S, and the analyticalresults using the approximations � 1 (black dots) and � 1 (black dashed line),for two values of the biomolecule layer thickness. Thickness of the biomolecule layer2 nm (left), and 0.1 nm (right).

In general it was seen that including the e�ect of volume in the model,thereby modelling the biomolecules as a surface charge and a volume insteadof a surface charge only, increases the charge induced in the FET uponattachment of biomolecules drastically.

The �rst application of the model is assuming that both the surfacecharge and the volume layer are not present before biomolecule attachment,and are present after biomolecule attachment. When comparing this withthe SCM in which no volume e�ect was included we found the following(EVM1). In the small potential approximation the charge induced in theFET upon biomolecule attachment is about 50 times larger in a model inwhich the biomolecules are modelled as a surface charge and a surface volumelayer than in a model in which the biomolecule is described as a surface


charge only. In the large potential approximation it was shown that the ratioof the results of the two models was dependent on the surface charge. Forlarger biomolecule surface charge the di�erence between the two models islarger. The error of ignoring the volume of the biomolecule when calculatingthe e�ect of attachment of biomolecules on the sensor can therefore be verylarge.

The second application of the model assumes that the biomoleculesare neutral. In this model the surface charge is already present beforebiomolecule attachment, and after biomolecule attachment both the sur-face charge, and the surface volume layer are present. It was seen that inthe small potential regime the e�ect of volume only was 50 times largerthan the e�ect of surface charge only, irrespective of the value of the surfacecharge. In the large potential regime the di�erence is even larger, and theratio is dependent on the value of the surface charge. The e�ect of a volumesurface layer is therefore tunable by tuning the surface charge of the sensorbefore biomolecule attachment.

4.2 The Excluded Volume Model for the cylindri-

cal sensor

Figure 4.10:

In this section the charge induced in the cylindrical FET is derived whenboth a surface charge and a dense biomolecule layer are located on the sensorsurface (see �gure 4.10).

As for the at sensor, a relation between the potential at the position band at position c is necessary. This is obtained by �rst �nding the electric�eld in the biomolecule layer with the use of a Gaussian surface in thebiomolecule layer, and integrating from b to c. This yields the relationbetween the two potentials,


Vc � Vb = ��FET + �S

2��0�r;bio= ln�cb

� ; (4.29)

or

c �b = �e

kBT

�FET + �S

2��0�r;bio= ln�cb

� : (4.30)

The relation between the charge induced in the FET and the surface poten-tial is given by the capacitance (equation (2.9)), inverting this relation weobtain

b =�cylFET

2�aCcyl

e

kBT: (4.31)

Inserting this into equation (4.30) and solving for �cylFET the following resultis obtained for the charge induced in the FET as a function of the potentialat position c

�cylFET = ��S +kBTe c 2��0�r;bio= ln

�cb

��0�r;bio

�aCcyl ln(c=b)+ 1

= ��S + �=� cylbioc

1 + � cylox =�cylbio

; (4.32)

where

� cylbio �ln�cb

��r;bio

; (4.33)

and � and � cylox are de�ned in equations (3.6) (3.22).

The expression for the charge in the double layer is given in equation(2.26). Multiplying by the circumference, 2�c, and noting that the doublelayer interface potential, 0, is c here, and that the radius of the cylindricalpotential,R, is c, we obtain the charge in the double layer per unit length ofthe FET

�DL = �2�� atc sinh

�c

2

�"1 +

( (�c))�2 � 1

cosh2�c

4

�#1=2

: (4.34)

Putting this together with equation 4.1 we obtain the equation that needsto be solved for c in order to obtain an expression for �cylFET,

� �S + �=� cylbioc


+ �S � 2�� atc sinh

�c

2

�"1 +

( (�c))�2 � 1

cosh2�c

4

�#1=2

= 0:

(4.35)


In �gure 4.11 the numerical solution of the above equation is shown as afunction of the surface charge for two di�erent thicknesses of the biomoleculelayer. It is seen that, if the surface charge is due to the charge of the attachedbiomolecules, it is reasonable to assume c � 1. If other charges are presenton the sensor surface, or when the FET is not grounded, the potential canbe much larger. Therefore the results for the approximation c � 1 are alsogiven.

Figure 4.11: The numerical solution of equation (4.35) is given as a function ofthe surface charge for di�erent values of the biolayer thickness. It is seen that c

stays well below zero. The values of the other variables are chosen such that theyare realistic for biosensing (see table at the end of chapter 2).


Assuming c � 1, equation (4.35) becomes

� �S + �=� cylbioc


+ �S � ��cylc c = 0; (4.36)

with

��cylc =�� atc

(�c): (4.37)

This equation has exactly the same form as the equation for the at geometry(equation (4.11)). The calculations are therefore also the same, and yieldfor the charge per unit length induced in the FET

�cylFET = � �S


1 +

� cylox =�cylbio

1 + ��cylc (� cylbio + � cylox )

!; (4.38)

or

�cylFET = � 1

1 + ��cylc � cylox

1 +

��cylc � cylbio

1 + 1=(��cylc � cylox ) + � cylbio=�cylox

!�S: (4.39)


The dependence on the variables is the same as for the at case, only thevariables themselves are di�erent. Also in this case the order of magnitudeof the variables is such that the second term in brackets is small comparedto the �rst term. The di�erence between the cylindrical results and the atresults will be discussed in section 4.3.

Since the values of the variables are not the same, some results are givenhere. For a cylindrical FET with standard surface charge, the charge in theFET in the EVM with a layer thickness of 10 nm is 93 times larger than thecharge in the FET in the SCM. the charge in the FET in the EVM with alayer thickness of 2 nm is 50 times larger than the charge in the FET in theSCM. the charge in the FET in the EVM with a layer thickness of 0.91 nmis 30 times larger than the charge in the FET in the SCM.


In the approximation c � 1 equation (4.35) becomes

� �S


+ �S = 0� �� atc exp

�c

2

�: (4.40)

which has the same form as equation (4.21). The same calculations can bedone, and all the results are the same as those in subsection 4.1.2. Also whencomparing the small and the large potential regime, the results of subsection4.1.3 can be used. The di�erence is only due to the fact that the variableshave di�erent values, and di�erent dependencies on the physical variables.The charge per unit length in the FET is

�cylFET = � �S�cylbio

� cylbio + � cylox

� 2�

� cylbio + � cylox

ln

"�S

�� atc

1

1 + � cylbio=�cylox

!#: (4.41)

In �gure 4.12the result of this approximation is shown together with theresult of the small potential approximation and the numerical result. Is isshown that, just like in the at geometry, the result of the small and thelarge potential approximation are very similar. The di�erence in the resultsof the at and the cylindrical FET are discussed in section 4.3.

4.3 Comparing the excluded volume e�ects in the

at and the cylindrical geometry

The dependence of the induced charges on the system parameters, � cylox and�� atc , and the variables, �S and � cylbio , is the same for the at as for thecylindrical geometry, both for the small potential regime (equations (4.13),(4.38)) as for the large potential regime (equations (4.23) and (4.41)). Thedi�erence is only due to the fact that these parameters and variables have


Figure 4.12: The numerical result of the charge induced in the at FET sensor,�FET (red line), as a function of the charge on its surface, �S, and the analyticalresults using the approximations � 1 (black dots) and � 1 (black dashedline), for two di�erent biomolecule layer thicknesses.

a di�erent dependence on the physical quantities, and therefore a di�erentvalue.

For the �rst application a plot of the charge induced in the at and inthe cylindrical FET is given in �gure 4.13 (a.) as a function of the surfacecharge, and in �gure 4.13 (b.) as a function of the biolayer thickness. Itis seen that more charge is induced in the at FET than in the cylindricalFET.

In �gure 4.13 (c.) the ratio of the EVM and the SCM result is shown as afunction of the biomolecule layer thickness tbio=b. The e�ect of the excludedvolume is seen to be large. Already for a small biomolecule layer thicknessthe induced charge in the FET is many times larger when a biomoleculelayer is present on top of the surface charge than when there is only asurface charge present. For both geometries this e�ect is substantial. It isseen that the error in considering only the charge, instead of considering acharge and a dense biomolecule layer, is larger in the cylindrical than in the at geometry.

For the second application the di�erence with the model without thevolume is considered. In �gure 4.13 (d.) this di�erence is plotted again as afunction of tbio=b. It is seen that the cylindrical geometry is more sensitiveto the presence of a surface layer.

It should be noted that setting the thickness of the biomolecule layersequal in the two geometries implies excluding more volume in the cylindricalgeometry. Therefore a comparison was made in which the excluded volumewas the same, and the layer thickness di�ers. For the at geometry 29 timesmore charge was induced in the FET in the Excluded Volume Model witha layer thickness of 0.94 nm than for the Surface Charge Model. In thecylindrical geometry with a layer thickness of 0.91 nm this factor was shown


to be 30. This is still more than in the at geometry. It can be concludedthat omitting the e�ect of the volume leads to a slightly larger error in thecylindrical case. It was seen before that screening had more e�ect in thecylindrical geometry than in the at geometry. The fact that the cylindricalgeometry is more sensitive to the volume e�ect, or that it is more sensitiveto the e�ect of removal of screening, is therefore an intuitive result.

The above results were obtained for the small potential regime, since forrealistic values of the surface charge, the sensor operates in this regime. Theresults for the large potential regime are similar, but all the values, fractionsand di�erences are larger.

In this chapter it was seen that the dense biomolecule has a strong e�ecton the induced charge in the FET, both for the at and the cylindrical sen-sor. The excluded volume could therefore very well be a sensing mechanismin biosensing. It should be noted, that the biomolecule layer will not becompletely dense in reality. Some ions will be present in the layer, and alsothe dielectric constant is likely to be somewhat larger than the dielectricconstant of the biomolecules. Further research is necessary to investigatethe charge induced in the FET when the biomolecule volume only partiallyoccupies the surface layer. A model of a uniform partial excluded volumelayer will be discussed in the next chapter.


a.)

b.)

c.)

d.)

Figure 4.13: The charge induced in the cylindrical FET sensor (red line), and inthe at sensor(green line) as a function of the charge on the FET surface, �S, andthe biomolecule layer thickness tbio=b. The values of the variables are taken to bethe standard values given at the end of chapter 2.


Chapter 5

The Partially Excluded

Volume Model

In the previous chapter the volume of the biomolecules was included in thecalculations as a dense layer. In a thin layer on top of the sensor surface theelectrolyte was completely replaced by biomolecules, removing the presenceof ions and changing the dielectric constant in that region to the value of thedielectric constant of the biomolecules. The picture sketched is a somewhatextreme situation, since the density of attached biomolecules depends on theconcentration of ions in the solution. Also between the biomolecules somespace needs to be present to allow attachment in the �rst place. There-fore another model is presented in this chapter, in which the volume of thebiomolecules is taken into account by considering a surface layer on the sen-sor in which only a fraction of the volume is occupied by the biomolecules(see �gure 5.1). In this surface layer only a fraction of the ions is pushedaway by the biomolecules, and the dielectric constant is replaced by a newdielectric constant, �r;new. This new dielectric constant will be taken to be acombination of the dielectric constant of the electrolyte, �r;w, and the dielec-tric constant of the biomolecules, �r;bio. This model is called the PartiallyExluded Volume Model (PEVM).

In this system there are four charges we need to take into account. Thecharge in the FET, �FET, the surface charge, �S, the ionic charge in thebiomolecule surface layer, �DL;I, and the charge in the electrolyte solution,�DL;II. The charge neutrality condition now is

�FET + �S + �DL;I + �DL;II = 0: (5.1)

The expressions for most of these charges were already derived in chapter2, only the expression for the charge in the surface layer was not derivedyet. For the at sensor this will be derived in this chapter, and the resultswill be calculated numerically. The cylindrical results will be derived usinga �nite element computer simulations (COMSOL).

77

78 THE PARTIALLY EXCLUDED VOLUME MODEL

Figure 5.1: The biomolecules exclude the electrolyte from part of the volume inthe surface layer. In the Partially Excluded Volume Model this is modelled by asurface layer with adjusted dielectric constant and ion concentration.

5.1 The partially Excluded Volume Model for the

at sensor

The at sensor is shown in �gure 5.2. Region I is the surface layer inwhich a fraction of the volume is occupied by the biomolecules, and theremaining volume by the electrolyte. Region II is taken as an electrolyteonly region, in which the biomolecule concentration can be neglected. Thepart of the volume that is occupied by the biomolecules contains no ionsand has a dielectric constant �r;bio. The part of the surface layer that isoccupied by the electrolyte has dielectric constant �r;w, and has the sameion concentration as the bulk electrolyte, only altered by the local potential.

Figure 5.2: The at biosensor in the Partially Excluded Volume Model. In thesurface layer part of the volume is occupied by biomolecules and part by the elec-trolyte.

To be able to model this layer as a uniform surface layer a new dielectricconstant, �r;new, is used, which can be calculated using the values of �r;bioand �r;w (equation (2.1) or (2.2)). Furthermore the discrete nature of the ion

THE PARTIALLY EXCLUDED VOLUME MODEL 79

concentration is removed by taking a new uniform charge density, �new(),which is only a part of the charge that would be present if the biomoleculesdid not exclude a certain fraction, f , of the volume �new() = (1� f)�().The dependence of the charge density on the ion concentration in the bulkelectrolyte, I0, is

�new() = (1� f)eI0 (exp [�]� exp [+]) : (5.2)

The charge density has a linear dependence on the ion concentration in thebulk. Taking only a certain fraction of the charge density, can be modelled astaking only a fraction of the ion concentration in the bulk, I0;new = (1�f)I0.The inverse Debye screening length in region I of the model, �I, is thereforeadapted in order to take this excluded volume e�ect into account, by takinga both a new dielectric constant and a new ion concentration.

The charge of the ions in both region I and II can be found by taking aGaussian surface enclosing both the biomolecule surface layer and the doublelayer in the electrolyte

�DL;I + �DL;II =�0�r;newkBT

e

dI(b+)

dx; (5.3)

with �DL;I, and �DL;II the surface charge per unit area of FET device inregion I and region II respectively, and I(x) the potential in region I. Thisgives the amount of charge in these two layers as a function of the electric�eld just at the outside of the sensor surface. The above expression showsthat the PEVM introduces an additional variable to the charge neutralitycondition (5.1). To �nd the derivative of the potential at position b in termsof the potential at position b and the potential at position c, we considerthe Poisson-Boltzmann equation that speci�es the potential in region I,

d2I(x)

dx2= �2I sinh

�I(x)

�: (5.4)

The left hand side can be rewritten to obtain

1

2

d

d

�dI(x)

dx

�2= �2I sinh

�I(x)

�: (5.5)

By integrating both sides over I from I(x) to I(c) we obtain

�dI

dx(c�)

�2��dI

dx(x)

�2= 2�2I (cosh((c

�))� cosh((x))): (5.6)

This can be solved for dI(x)=dx to get

dI

dx(x) = �

s�2�21(cosh((c�))� cosh((x))) +

�dI

dx(c�)

�2: (5.7)


At position c there is no surface charge present, so the electric �elds in regionI and II are related the following way

dI

dx(c�) =

�r;w�r;new

dII

dx(c+): (5.8)

We also know that

dII

dx(c+) = �2�II sinh

�c

2

�; (5.9)

since the result for the double layer charge from chapter 2 is valid in region II(see equation (2.18)). Combining equations (5.7), (5.8), and (5.9) we obtain[10]

dI

dx(x) = �

s�2�2I (cosh(c)� cosh((x))) +

�2�II�r;w�r;new

�2sinh2

�c

2

�

= �2�I

vuutsinh2�(x)

2

�+

��II�r;w�I�r;new

�2� 1

!sinh2

�c

2

�

= �2�Issinh2

�(x)

2

�+ C; (5.10)

where

C = C(c) =

��II�r;w�I�r;new

�2� 1

!sinh2

�c

2

�: (5.11)

Choosing x = b and substituting the result of equation (5.10) into equation(5.3) we obtain an expression which is only dependent on the potentials atpositions b and c,

�DL;I + �DL;II = (� sgnb)2�0�r;newkBT�I

e

ssinh2

�b

2

�+ C(c): (5.12)

In the above equation the sign has been �xed. Since we know that thepotential in the bulk of the electrolyte is zero, the charges with sign oppositeto that of the potential, b, will ow towards the sensor. Therefore the totalnet charge in the double layer, �DL;I + �DL;II, must have a sign opposite tothat of b. This has been indicated with �(sgnb). Throughout this thesisthe potential is positive, since the surface charge was taken to be positive,therefore the negative sign is appropriate here. It can be seen that forf = 0, or �I�r;new = �II�r;w, the above result becomes equal to the result ofthe charge in the double layer without an excluded volume (equation 3.3).


It can also be shown that in the limit �I = 0, or f = 1, the result for theexcluded volume model is retrieved (equation (4.9)).

Equation (5.12) is dependent on both b and c. Therefore also equation(5.1) is dependent on both b and c

�FET(b) + �S + (�DL;I + �DL;II)(b;c) = 0: (5.13)

In order to solve the problem, a relationship between b and c has to bederived. This can be obtained by rewriting equation (5.10), and performingthe integration

Z c

bdx = (� sgnb)

Z I(c)

I(b)

dI

2�I

rsinh2

�I

2

�+ C

; (5.14)

which is equal to

c� b =1

(�(sgnb))2�IpC

Z c

b

dIr1 + 1

C sinh2�I

2

� (5.15)

This equation provides the relation between b and c. Recall that theconstant C also depends on c.

Using this result, for a given value of b, the corresponding value of c

can be found numerically. The values of b and c can be used together withequation (5.12), to �nd the total net double layer charge for that speci�cvalue of b. Doing this for many values of b, a plot can be made of thetotal double layer charge as a function of b, this is shown in �gure 5.3.Plotting ��FET(b) � �S in the same �gure, the intersection tells us whatthe solution is, or for what value of �FET the equation

�DL;I + �DL;II = �(�FET + �S) (5.16)

is satis�ed (equation (5.1)). The surface charge is multiplied with 2�b inorder to get the surface charge per unit length of the device, which is thequantity shown in the plot.

This is a cumbersome method, and the results will be analyzed furtherwith the use of elliptic functions. The elliptic integral of the �rst kind isde�ned as

F (�;m) �Z �

0

d�p1�m sin2 �

= u; (5.17)

and its inverse is the Jacobian Amplitude

Am(u;m) = �: (5.18)


Figure 5.3: The solid line indicates the charge of the ions both in region I and inregion II of the at FET, �DL;I+�DL;II, as a function of the sensor surface potential,b. The dashed line shows the negative of the charge induced in the FET and thecharge on the surface, ��FET(b)� �S. The intersection shows the solution of theproblem.

To rewrite equation (5.15) in terms of elliptic integrals a few steps haveto be made. We start by de�ning the following function

G(�;m) �Z �

0

d�p1 +m sinh2 �

; (5.19)

which obeys the following relation

G(�;m) =1pmG

��;

1

m

�; (5.20)

with � = arcsinh(pm sinh�). If the value of m is larger than one, this

relation can be used to obtain the same function where the prefactor of thesquare of the hyperbolic sine is smaller than 1. Once a function with aprefactor smaller than one is obtained the following relation can be proven

G(�;m) = F (�; 1�m); (5.21)

where � = arctan(sinh�), and m < 1 (see appendix B).

Taking a function G(�;m), with m > 1, and using the two above equa-tions we obtain

G(�;m) =1pmF

�arctan(

pm sinh�); 1� 1

m

�: (5.22)

With this relation equation (5.15) can be rewritten in terms of Ellipticintegrals if C < 1. This is true for the physical variables that are realisticfor biosensing. For C > 1 equation (5.21) can be used immediately.


c� b =1

�(sgnb)�IpC

�G

�c

2;1

C

��G

�b

2;1

C

��

=1

�(sgnb)�I

�F

�arctan

�1pCsinh

�c

2

��; 1� C

�

�F�arctan

�1pCsinh

�b

2

��; 1� C

��: (5.23)

This can be inverted with the use of the Jacobian amplitude to obtain therelation between b and c

b = 2arcsinh

pC tan

"Am

(F

arctan

�1pCsinh

�c

2

��; 1� C

!

+(c� b)(sgnb)�I; 1� C

)#!: (5.24)

Using this relation we can eliminate b from equation (5.13). This yieldsan equation with only one unknown, and the problem can be solved. Sothe above result combined with the results for �FET(b) (equation (2.9)),(�DL;I+�DL;II)(b;c) (equation (5.12)), the value of the surface charge, �S,and equation (5.13), can be used to calculate the value of �FET numerically.

In �gure 5.4 the numerical solution of the charge in the FET is given asa function of the excluded volume fraction, f . To be able to compare thecharge induced in the at FET with that induced in the cylindrical FET,the results are multiplied with 2�b to obtain the charges per unit length ofthe device. For f = 0 the �gures show the result when the volume of thebiomolecules has not been taken into account, which yields the result of theSCM. For f = 1 the surface layer is a dense layer, here the �gure shows theresult of the EVM.

Figure 5.4 (a.) shows two di�erent lines. The bottom line shows theresult in case the biomolecules lie at down on the surface. The surfacelayer thickness is the diameter of the biomolecules, which was taken to be2 nm. The dielectric constant of the surface layer changes with f accordingto equation (2.2). The top line in �gure 5.4 shows the results assumingthat the biomolecules are oriented perpendicular to the sensor surface. Thein uenced surface layer thickness is the length of the biomolecules, whichwas assumed to be 10 nm. Here equation (2.1) is used to �nd the dielectricconstant.

The dependence on f is di�erent for the two biomolecule orientations.This is caused by the fact that the dielectric constant has a di�erent depen-dence on f for the two orientations (see �gure 2.11). Since �I is dependenton the dielectric constant, also this parameter has a di�erent dependence


on f for the two orientations. It is seen that for both orientations for largeexcluded volume fractions, f , the charge induced in the FET changes veryrapidly. Such large values of f are not realistic, however. For smaller valuesof f the induced charge changes more rapidly with f when the molecules lie at on the surface. It should be noted that in addition to this e�ect, also thevalue of f is di�erent for the two orientations. When the biomolecules areoriented at on the sensor surface the value of f was shown to be about 0.5.In case the biomolecules are positioned straight on the sensor surface thevalue is about f = 0:1. This increases the di�erence in the charge inducedin the FET for the two orientations even more.

a.) b.)

Figure 5.4: (a.) The dependence of the induced charge per unit length in the FETin the case the biomolecules lie at on the sensor surface (bottom line, tbio = 2nm)and in the case the biomolecules stand straight on the surface (top line, tbio =10nm), for di�erent surface layer thicknesses. (b.) The ratio of this result and theSCM result.

There are di�erent applications of this result. In the �rst application(PEVM1) we want to �nd what the error is of modelling the biomolecule asa surface charge, ignoring any e�ects of volume. The expression of interestis

��PEVM1FET

��SCMFET

=�PEVMFET

�SCMFET

: (5.25)

Note that the result of the SCM is independent of volume, so independentof the surface layer thickness, or the excluded volume fraction.

In �gure 5.4 (b.) again both the result assuming the biomolecules areoriented at on the sensor surface (line at the top), and that assuming thebiomolecules stand up straight (line at the bottom) are shown. There aretwo di�erent points to distinguish in this �gure.

The �rst point of interest lies on the bottom line, corresponding to thestraight orientation, and f = 0:1. The induced charge per unit length ofdevice is 1.1 times than that of a model without volume taken into account(or f = 0). The second point lies on the curve on top corresponding to the at orientation, and f = 0:5. For this value of the excluded volume fractionthe induced charge per unit length in this model is 5.6 times that of themodel where the e�ect of volume was not taken into account.


Not taking into account the e�ect of volume gives an underestimationof the induced charge, which is larger if the biomolecules lie at on the sen-sor surface than when they stand straight. It is expected that in realitythe orientation is not at or straight, but somewhere in between. The re-sult is therefore also expected also to be in between the results of the twoorientations.

In the second application (PEVM2) we compare the result of modellingthe biomolecule as a surface charge only, with the result of modelling it asa volume only. The expression of interest is

��PEVM2FET

��SCMFET

=�PEVMFET � �SCMFET

�SCMFET

=�PEVMFET

�SCMFET

� 1: (5.26)

This is the same expression as for the �rst application, only with a valueof 1 subtracted. It is seen that the induced charge in the PEVM in thesecond interpretation is 4.6 and 0.1 times that of the SCM, for the case thebiomolecules lie at and stand up straight respectively. It should be noted,however, that these results were derived when the device was not especiallydesigned for volume detection. When the surface charge is increased inorder to increase the in uence of the volume of the biomolecules, this factorbecomes larger. This way the sensor can be designed to be more sensitiveto the presence of a surface charge layer.

The dependence on the biomolecule layer thickness is shown in �gure5.5. The value of the charge induced in the FET is shown at di�erentvalues of the excluded volume fraction. It is seen that the in uence of thebiolayer thickness is large for small biolayer thicknesses. For larger biolayerthicknesses the value of the induced charge in the FET stabilizes. Both inthe case that the biomolecules lie down and in the case the biomoleculesstand up straight, the value of the biolayer thickness is in this stabilizedregime for a realistic biolayer thickness (10 nm for top line, 2 nm for bottomline).

This can be understood as follows. If all the �elds are screened withinthe surface layer, then there are no �elds present in the electrolyte solution(region II). The charge density of the electrolyte is zero in the regions wherethere are no �elds present. Therefore, replacing part of the electrolyte inthis region with neutral biomolecules does not change the charge density inthis region. Also the change in the dielectric constant of this region doesin uence the problem since no �elds are present in this region. Extendingthe thickness of the biomolecule surface layer in this case therefore has noin uence on the charge induced in the FET.


The distance from the sensor surface at which all the �elds are screenedis characterized by the screening length in that layer. This is the screeninglength, ��1I , with an adjusted ion concentration and dielectric constant. Incase the biomolecules are oriented parallel to the sensor surface his lengthis about 0.3 nm. When the biomolecules are oriented perpendicular to thesensor surface this is about 1 nm. At distances larger than this adjustedscreening length, the value of the induced charge in the FET is the same asthe value of the charge induced in the FET for in�nite layer thickness. It canbe concluded that the results of the PEVM are then the same as the result ofthe SCM, with adjusted dielectric constant and adjusted ion concentration.This means that the less complex, and more insightful formulas of chapter3 can be used to describe partial occupation of the surface layer by thebiomolecules.

The fact that the system is in the stable regime also means that thereis no di�erence in the sensitivity of the sensor to a biomolecule with di-mensions as discussed here and a biomolecule that is twice as long. Abiomolecule with a larger diameter, one that lies at on the sensor surfaceor one that is curled up, will give a larger excluded volume fraction andtherefor a larger sensitivity. In order to increase the ease of detection of acertain biomolecule therefore one could try to attach additional groups tothe biomolecule of interest that increase the volume of the biomolecule, orin uence the orientation of the biomolecule.

Figure 5.5: The charge induced in the FET in the PEVM as a function of thesurface layer thickness. The bottom line gives the result for an excluded volumefraction of f = 0:5. A realistic value of the surface layer thickness is tbio = 2 nm.In the top line the excluded volume fraction was taken to be that of biomoleculesstanding up straight, f = 0:1, and the realistic surface layer thickness is tbio = 10nm.

When considering the in uence of the ion concentration on the volumee�ect we look at �gure 5.6. We compare three di�erent cases with the sametotal excluded volume. The �rst case is a result of the EVM, where the layerthickness is 0.94 nm and the excluded volume fraction is 1. The second caseis the result when the biomolecules lie down, giving a layer thickness of 2nm and an excluded volume fraction of 0.5. The third case is the result forbiomolecules that stand straight on the sensor surface, corresponding to a


surface layer thickness of 10 nm and an excluded volume fraction of 0.1. Theblack dashed line corresponds to the SCM result where no volume e�ect istaken into account. In �gure 5.6 (a.) the total charge in the FET is shown.It is seen that screening has a negative in uence on the charge inducedin the FET. In the �rst application (PEVM1), where the biomolecules aremodelled as having both a charge and a volume, it is therefore bene�cial tohave a small ion concentration.

In the second application (PEVM2) the sensor surface itself is chargeddue to for example oxide charge, and the biomolecules are neutral. Thecharge induced in the FET upon biomolecule attachment in this model isshown in �gure 5.6 (b.). The EVM result is the line at the bottom. Fora larger ion concentration the di�erence between the EVM and the SCMincreases. When looking at the results of the PEVM (middle and top line),however, the opposite behavior is shown. This is due to the fact that for verylarge ion concentrations only a small available volume for the ions is alreadylarge enough for the ions to completely dominate the system. Almost all ofthe surface charge is compensated by the ions and the charge induced in theFET is small.

In �gure (c.) the top line of �gure (b.) is shown again, showing thatthere is an optimum value of the ion concentration. For biosensing, the ionconcentration is not a variable, and the ion concentration imposed by thesystem is not likely to be at this optimum. When comparing these resultsto those of the SCM (�gure 5.6 (d.)), we see that the charge induced inthe FET upon attachment of neutral biomolecules with a volume is muchlarger than the charge in the FET in a model without an excluded volume.It is seen that although the e�ect of the volume decreases for increasing ionconcentration, the ratio of this result and the SCM result increases. Forlarger ion concentrations therefore the presence of the biomolecule volumeis easier to detect than the presence of the biomolecule surface charge.

Adding 1 to the result in �gure 5.6 (d.) we obtain �PEVMFET =�SCMFET =��PEVM1

FET =��SCMFET . The error in modelling the biomolecules as a chargelayer only, without taking into account the e�ect of the volume, is seen tobe larger for larger ion concentrations. For biosensing the ion concentrationis large, therefore it is important that the e�ect of the volume is taken intoaccount.

The dependence of the excluded volume e�ect on the surface charge,�S, is shown in �gure 5.7. The results are shown both for the case theSCM is in the small potential regime (a.) and for the case it is in the largepotential regime (b.). It is seen that increasing the surface charge increasesthe excluded volume e�ect.

This can be understood the following way. For increasing surface charge,�S, the surface potential increases, which increases the charge density of theelectrolyte close to the sensor surface. Replacing part of this charged elec-trolyte solution with neutral biomolecules changes the charge distribution


a.) b.)

c.) d.)

Figure 5.6: (a.) The total charge in the FET as a function of the ion concentrationin the bulk electrolyte for three di�erent cases, with equal amounts of excludedvolume. The line at the bottom corresponds to a dense surface layer, the middle linecorresponds to the situation where the biomolecules are oriented perpendicularlyto the sensor surface, and the line at the top corresponds to the situation that thebiomolecules are oriented parallel to the sensor surface. The black dashed line isthe result when no biomolecule volume is present in the model, i.e. the SCM result.(b.) The di�erence of the three before mentioned cases and the SCM result. (c.)The top line from (b.) shown on a smaller scale. (d.) The ratio of the result of thesecond application of the PEVM and the SCM.

in the vicinity of the sensor. The larger the charge density in the electrolyteclose to the sensor surface, the larger the change in the charge distributionin the vicinity of the sensor upon biomolecule attachment. This causes againa larger change of the charge induced in the FET.

a.) b.)

Figure 5.7: The charge induced in the FET upon attachment of neutralbiomolecules as a function of the surface charge, in the small potential regime(a.) and in the large potential regime (b.).


5.2 The Partially Excluded Volume Model for the

cylindrical sensor

The cylindrical geometry as discussed in this section is shown in �gure 5.8.Just like in the at model the surface layer is now modeled as a layer inwhich part of the volume is occupied by the electrolyte solution, and partby the biomolecules. The layer is taken to be a uniform surface layer withdielectric constant �r;new, and adjusted screening length �I. Both �r;new and�I are dependent on the excluded volume fraction, f . The problem can notbe solved analytically. Therefore the results presented in this section werederived using a �nite element method (COMSOL).

In �gure 5.10 (a.) the charge induced in the cylindrical FET is shownas a function of the biomolecule layer thickness, both for the case that thebiomolecules are oriented at on the sensor surface and for the case thatthey stand straight on the sensor surface. It is seen that, just as for the at case, the charge induced in the FET is larger when the biomolecules lie at on the sensor surface and the values of the charges induced in the FETstabilize. This implies that for biomolecule layer thicknesses larger than thedistance of stabilization, the results of the SCM can be applied, reducingthe complexity of the calculations.

Figure 5.8: Schematic of the nanowire biosensor in the PEVM. The biomoleculesexclude the electrolyte from a certain fraction of the volume in the surface layer.

It should be noted that, in contract to the at geometry, the excludedvolume fraction in the cylindrical geometry varies with distance, f = f(r),when modeling the biomolecules as rigid rods(see �gure 5.9). This impliesthat also the dielectric constant and the screening length vary with distance.Taking the layer to be a uniform layer, with a �xed excluded volume fraction,leads to excluding a larger total volume in the cylindrical geometry thanin the at geometry. In order to account for the fact that the sensor iscylindrical, the the excluded volume fraction at the surface is the value ofthe excluded volume fraction of the uniform surface layer, f(b) = f , whilein the remainder of the biomolecule layer the excluded volume fraction is


calculated as follows:

f(r) = fb

r: (5.27)

The results of this distance dependent partially excluded volume model (dis-tance dependent PEVM) are presented in �gure 5.10 (b.). It is seen that theresult also stabilizes in this model for layer thickness larger than a couple ofnanometers.

Figure 5.9: Schematic of the cylindrical nanowire biosensor in the PEVM. Thefraction of the volume occupied by the biomolecules is largest close to the surfaceof the sensor.

It is expected that for small values of the biolayer thickness the di�erencebetween the cylindrical uniform PEVM and the cylindrical distance depen-dent PEVM is small, since the di�erence in the value of the excluded volumefraction f is in that case small. In �gures 5.10 (c.) and (d.) this can indeedbe seen. For larger biolayer thicknesses we see that the di�erence betweenthe curves increases. Due to stabilization of the curves, at a certain distancefrom the sensor surface, the di�erence between the two models remains thesame.

Depending on the biological system of interest, the �rst or the secondmodel described here gives more realistic results.

In the next section the second model, the distance dependent PEVM,will be used to compare the at and the cylindrical sensor. This was chosenbecause the total excluded volume is the same in the cylindrical distancedependent PEVM as in the at distance dependent PEVM.

5.3 Comparing the e�ects of a partially excluded

surface layer for the at and the cylindrical

geometry

In this section the at and the cylindrical geometry will be compared inthe PEVM. To be able to compare the cylindrical results with the resultsof the at sensor, the value f is used with a somewhat di�erent meaning.Before it was the fraction of the volume of the biomolecule surface layer that


a.) b.)

c.) d.)

Figure 5.10: The charge induced in the cylindrical FET as a function of thebiomolecule layer thickness for (a.) a uniform surface layer, (b.) a surface layer withan excluded volume fraction dependent on the distance to the surface, f = f(r).These results are compared in (c.) for the case that the biomolecules are orientedperpendicular to the sensor surface, and in (d.) for the case the biomolecules areoriented parallel to the sensor surface.

was occupied by biomolecules, here it is the fraction of the sensor surfaceoccupied by biomolecules. In the case of the at sensor this is equal tothe excluded volume fraction, in the cylindrical case this is only equal tothe excluded volume fraction at the sensor surface. Comparing the atand the cylindrical results with the same value of f then implies comparingthe results for the same surface density of biomolecules and the same totalexcluded volume.

In �gure 5.11 (a.) the charge induced in the FET is shown as a functionof the excluded volume fraction for both the cylindrical (black dots) and the at sensor (green line), and for the two biomolecule orientations. It is seenthat, according to the PEVM, more charge is induced in the at sensor thanin the cylindrical sensor just like in the SCM. When dividing the results bythe SCM result we obtain the results in �gure 5.11 (b.). Since the result ofthe SCM was di�erent for the two geometries, it is seen that the error inmodelling the sensor without volume, for realistic values of f , is comparablefor the at and for the cylindrical sensor.


a.) b.)

Figure 5.11: (a.) The charge induced in the FET is shown as a function of theexcluded volume fraction for both the cylindrical (black dots) and the at sensor(green line). The upper line and dots are the results when the biomolecules areoriented perpendicular to the sensor surface, the lower line and dots are the resultswhen the biomolecules are oriented parallel to the sensor surface. (b) The result in(a.) divided by the SCM result.

Chapter 6

Discussion and Conclusion

The question that raised interest to start this thesis was: How is it possiblethat nanowire biosensors give such good results in the lab, while in theoryscreening is expected to prohibit this? This was the starting point for thisthesis. Di�erent steps are taken to achieve better understanding of thesensing mechanism.

First a simpli�ed metallic nanowire sensor was studied, where the biomo-lecules were modelled as a surface charge. As predicted by many others, itwas found that screening reduced the sensitivity of the sensor. It was alsofound that the dielectric constant of the sample solution has a large e�ect. Alarge dielectric constant decreases the sensor sensitivity. In many biologicalsamples the solution is based on water, with a large dielectric constant.These solutions contain a large amount of ions, which is necessary for thestability and attachment of the biomolecules. The ion concentration andthe dielectric constant of the biological sample cannot be chosen, but aredictated by the biological system of interest. Both the ion concentrationand the dielectric constant of the solution close to the sensor surface can bein uenced by the presence of biomolecules. The biomolecule volume, thatis usually ignored, can therefore have a large e�ect on the sensitivity. Thequestion was raised whether this surface charge model described the realsystem good enough.

To test whether the biomolecule volume indeed has a considerable e�ecta model was studied in which a thin layer on the sensor surface was occupiedby biomolecules. This excludes the ions from this region and the dielectricconstant of the layer is taken equal to the value of the dielectric constantof the biomolecules. It was shown that in a model where the biomoleculesare modelled as a surface charge and a surface volume, instead of only asurface charge, the charge induced in the sensor increases greatly. Roughestimations show this can be as large as a factor 80. The problem of screeningis shown to be reduced due to the biomolecule volume. If the e�ect ofvolume is not taken into account in the calculations this can lead to a strong

93

94 DISCUSSION AND CONCLUSION

underestimation of the sensor sensitivity. This study showed that even aneutral biomolecule layer can be detected by the nanowire sensor. Thee�ect of this neutral biomolecule layer exceeds many times the results ofthe biomolecule surface charge. The screening due to the presence of ionsis not actually a problem when considering the biomolecule volume, but anecessity for the volume e�ect.

These results showed that the volume of the biomolecules can have a largein uence on the detection of the molecules. However, a uniform biomoleculelayer excluding all ions is not that realistic. Therefore the surface layer wasconsidered being partially occupied by biomolecules. It was shown that theresults depend greatly on the surface layer characteristics. The orientation ofthe biomolecules determines the fraction of volume occupation at a certaindistance from the sensor surface, which in turn determines the dielectricconstant.

Understanding this large in uence of the biomolecule volume, allowsfor �ne tuning of the sensor. Adding a �xed surface charge to the sensorsurface for example increases the volume e�ect. Also one could think aboutattaching groups to the biomolecules of interest to increase their volume,or making sure the biomolecules lie at on the sensor surface. Anothermore complex use of this volume understanding can be depositing a certainmaterial on the sensor after biomolecule attachment.

The strong dependence on the precise characteristics of the biomoleculevolume suggests that a model in which the biomolecule layer is taken to be auniform layer probably will not yield correct predictions. It is recommendedthat the system is studied further, taking into account the discreteness ofthe biomolecules.

The before mentioned studies have been done both for a cylindricalnanowire sensor, and for a at sensor. The cylindrical geometry is moresimilar to the realistic geometry, but the at sensor was included since thecalculations are more insightful in this geometry. Both the at and thecylindrical sensor gave similar results. It was shown that the models forboth geometries can be described using the same equations, with a di�erentdependence of the variables on the physical quantities. The results showedthat the cylindrical sensor induced less charge than the at sensor. This doesnot imply that the at sensor is more sensitive than the cylindrical sensor,since the discreteness of the biomolecules was not taken into account. Pinch-ing for example is only possible in the cylindrical geometry, and can resultin single molecule detection.

The material of the sensors was taken to be metal and not a dielectric.If one is interested in including dielectric properties, the simple capacitancethat was used in this thesis can be replaced with the capacitance of a di-electric sensor.

The values of the charges of the biomolecules and their attachment to thesensor that were used were rough estimations. For each biomolecule system

DISCUSSION AND CONCLUSION 95

this will be di�erent. Depending on the local potential the attachment ofbiomolecules and the local ion concentration varies. This in turn can havean in uence on the charge of the biomolecules and the sensor surface. Thispotential dependence was not taken into account in the models described inthis thesis.

To conclude, we can answer the question: Is the in uence of volumeimportant for biosensing or is it negligible compared to the e�ect of thecharge of the biomolecules? It was shown that the e�ect of volume cannot be neglected, and in all future studies this topic should be addressed.This can explain why nanowire biosensing experiments show a much highersensitivity than one would expect from simple charge calculations.

96 DISCUSSION AND CONCLUSION

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Sensing Mechanism of Nanowire Biosensors van der Lans.pdf · In chapter 3 only the in uence of the charge of the biomolecules is stud-ied. In chapters 4, and 5 the e ect of volume