Eindhoven University of Technology MASTER The development of a low temperature MFM Meertens, Arjan

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Eindhoven University of Technology

MASTER

The development of a low temperature MFM

Meertens, Arjan

Award date:

2009

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The development of a low temperature MFM

Arjan Meertens Master thesis May 2008 - June 2009

Under supervision of:

Ir. N.A.J.M. Kleemans Prof. dr. P.M. Koenraad Department of Applied Physics

Photonics and Semiconductor Nanophysics University of Technology Eindhoven

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Abstract

In order to image the magnetic contrast of a magnetic semiconductor structure with a Curie temperature well below room temperature a low temperature Magnetic Force Microscope (MFM) is developed. As a first step Atomic Force Microscope (AFM) measurements and calibrations have been performed at room temperature, liquid nitrogen temperature (77K) and liquid helium temperature (4K). The piezo scanners show a large nonlinearity that differs for the three temperatures. The images need to be corrected for these non-linearities in order to get the correct image sizes. In AFM mode 4nm high features can be distinguished in clustered groups of quantum dots at low temperature. The higher Q-factor of the cantilever at lower temperature has a strong effect on the imaging modes that we have studied. This project ended successfully as we have were able to perform high resolution MFM imaging at 4K.

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Introduction

The Atomic Force Microscope, AFM, is a member of the family of scanning probe microscopes. Scanning probe microscopes use a probe that scans the sample surface. An image of the surface is then obtained by moving the probe in a raster scan over the surface while the probe-surface interaction is monitored.

The first AFM was invented in 1986 by Binnig, Quate and Gerber [1]. The probe, also called tip, used in AFM measurements is at the end of a Silicon cantilever. When the tip is brought in close proximity to the sample, forces between the tip and the sample cause the cantilever to deflect. Depending on the mode of operation of the AFM different aspects of the cantilever can be monitored. The most common mode for AFM is the lock-in mode. In this mode the cantilever is driven near its resonance frequency and the amplitude and phase of the tip are monitored, which are used to make a topography image of the surface. But there are other interesting aspects of surfaces that can be measured. The magnetic contrast of the surface can be imaged with a Magnetic Force Microscope, MFM. Here the AFM-tip is coated with a ferromagnetic layer in order to detect the magnetic field of the sample.

Magnetically doped semiconductors have great potential in spintronic de- vices. GaMnAs is such a semiconductor where Mn acts as the magnetic dopant.

Since the Curie temperature for these structures is below room temperature [2], low temperature measurements have to be performed in order to investigate their magnetic properties. An MFM that is operational at low temperatures is capable of performing such a task. This thesis describes the development of a low temperature MFM operational at 4K, 77K and 293K.

In the first chapter the theory behind the AFM is discussed. First the interferometer technique, used as a detection mechanism for the deflection of the cantilever, is discussed. Next the forces working between the tip and the sample and the cantilever deflections caused by these forces are discussed. Chapter 1 is concluded with the different modes that can be used to operate the AFM. Chap- ter 2 describes the setup used to perform the low temperature AFM and MFM measurements. In chapter 3 we show the results of these measurements, and more importantly, the feasibility of using this setup to perform low temperature MFM.

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Chapter 2

Theory

The atomic force microscope (AFM) is a microscope to image the topography of a sample with (sub)nanometer precision. The AFM consists of a cantilever with a sharp tip at its end. When the tip is brought in close proximity to the sample, forces between tip and sample lead to a bending of the cantilever. In figure 2.1 the bending of the cantilever due to tip-sample forces is depicted. The cantilever bending is a direct measure for the local height of the sample. A fiber is mounted near the back of the cantilever and a laser beam leaves the fiber and reflects on the back side of the cantilever back into the fiber. If the cantilever is bended the fiber-cantilever distance is different than for the unbended cantilever.

This path difference can be measured with an optical interferometer as will be explained in section 2.1.

Laser Output, Fiber End

Tip

Sample

(a) (b)

Cantilever

Figure 2.1: A schematic representation of the interferometric AFM. (a) Un- bended cantilever near the sample surface. (b) Bended cantilever near the sample surface. The fiber-cantilever distance is different for the two cantilevers.

In this chapter we will first discuss the optical interferometric technique.

Next the forces working on the cantilever will be discussed followed by the reaction of the cantilever on these forces. The reaction of the cantilever is split in two parts, first the cantilever dynamics will be considered in static mode, after that in oscillation mode. This chapter is concluded with information about the different modes that can be used to operate an AFM.

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2.1 The interferometer technique

There are several methods to detect the deflection of the cantilever. The most commonly used detection method for AFM is the beam deflection technique. In this technique the cantilever displacement is measured by detecting the deflection of a laser beam reflected on the back side of the cantilever with a quadrant detector. This can be seen in figure 2.2 (a). Another technique is shown in figure 2.2 (b). This is the capacitance technique in which a small plate is mounted above the tip so the plate and cantilever form a capacitor. The capacity is a measure for the deflection of the cantilever. A third method uses a piezoelectric cantilever. When the cantilever bends it will produce an electric field which can be measured by a current detector as is shown in figure 2.2 (c).

I

(a) (b) ( ) c

Figure 2.2: Three cantilever deflection detection methods. (a) The beam deflection technique with the quadrant detector above the cantilever. (b) The capacitance technique. (c) The technique with the piezoelectric cantilever.

The AttoAFM-I uses the interferometric technique, see figure 2.3. In theory this technique gives an equal signal to noise ratio as the commonly used beam deflection technique [3] but the beam deflection technique is easier to use because it has less optical components. The main problem with the beam deflection technique is that the detector has to be close to the cantilever and thus to the sample. Because our setup is operational at low temperatures it is practical to use the interferometric technique. Here the detector can be placed far away from the cantilever and does not have to be cooled down and operated in a magnetic field.

The amount of light that reaches the detector depends on the distance between the fiber and the cantilever because of the interference between the light that reflects from the cantilever and from the fiber-air interface, see figure 2.3.

The total electric field of the detected light consists of two parts and is given by equation 2.1, the first part is the light that reflected at the air-fiber interface and the second part is reflected from the cantilever.

Et= rfE0+ rct²_fE0cosθ (2.1) Here Et is the total electric field going to the detector, E0 is the electric field of the original laser signal, rf and tf are the reflection and transmission coefficient of the end of the fiber and rc is the reflection coefficient of the cantilever. θ is the phase difference between the two different parts of the signal and is given by equation 2.2. This phase difference gives rise to the interference.

θ = 4πd

λ (2.2)

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E0 r Ef 0

t Ef 0 r t Ec f 0

r t E cosc f² 0 q

Figure 2.3: Schematic side view of the fiber end on top of the cantilever. The incoming light gets reflected by the fiber-air interface as well as by the cantilever.

The reflections give rise to the interferometric signal.

Here d is the distance between the fiber and the cantilever and λ is the wave- length of the laser light. The detector measures the power, therefore we rewrite the equation in terms of the irradiance I, the radiant power density. With I = ǫ0chE²i this gives the following equation.

It= ǫ0chEt2

i = I¹+ I2+ 2p

I1I2cos θ = ǫ0cE₀²(r²_f+ r²_ct⁴_f+ 2rfrctf2cos θ) (2.3) here It is the total irradiance, I1 is the irradiance of the beam reflected by the fiber-air interface and I2is the irradiance of the beam reflected by the cantilever.

The sum of the reflection and transmission coefficient of the irradiance should be equal to one. Therefore we can introduce new irradiance reflection and transmission coefficients to simplify things. Rf = r²_f = (1 − t²f) and Rc = r²_c which leads to:

It= ǫ0cE₀²(Rf+ Rc(1 − Rf)²+ 2(1 − Rf)p

RfRccosθ) (2.4) The reflectivity of the fiber can be calculated if we know the indices of refraction of air and the fiber [4]. nair ≈ 1 and nf iber ≈ 1.5, which gives:

Rf = (nf iber− nair

nf iber+ nair) ≈ 0.04 (2.5)

.

So Rf << 1 and we can simplify 2.4 to:

It≈ ǫ⁰cE₀²(Rf+ Rc+ 2p

RfRccos θ) (2.6)

The visibility of an interference signal is determined by contrast between the maximal and minimal irradiance. If a cantilever starts in an equilibrium position it can deflect up or down depending on the force working on it. To be able to measure both up and down deflections the cantilever should be positioned in the middle between the maximum and minimum where cos θ = 0. This means d = ^nλ₈ . This is also the position that has the highest sensitivity for deflections and thus changes in d. If the cantilever starts at this equilibrium position the visibility is given by:

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v =Imax− Imin

Imax+ Imin

= 2p RfRc

Rf+ Rc

(2.7) Now we can call Ieqthe irradiance at equilibrium, Ieq =^I^max^+I₂ ^min = ǫ0cE₀²(Rf+ Rc) and we can write Itin its simplest form:

It= Ieq(1 − v cos θ) (2.8)

This is a good approximation when the distance between the fiber and the cantilever is large. For small distances the interference is not simply build up out of the two reflected signals but also higher order reflections between cantilever and fiber need to be taken into account as is shown schematically in figure 2.4.

Figure 2.4: Side view of the cantilever and the fiber end. The arrows represent the path of the light. For small cantilever-fiber distances higher order reflections have to be taken into account.

The total contribution to the interference signal by light reflected from the cantilever and the fiber is not constant, but depends on the distance between the fiber and the cantilever. Equation 2.1 now changes into a more complicated form [5].

Et= rfE0+ X∞ n=1

t²_frc(rfrc)ⁿ⁻¹E0V cos(ϕ) (2.9) ϕ is the phase difference between all the interference signals and is given by equation 2.10.

ϕ = 4πdn

λ + π(2n − 1) (2.10)

V is the loss due to the divergence of the laser beam after leaving the fiber and is also dependents on d. It is the area where the light can enter the fiber again divided by the area of the diverged laser beam as shown in figure 2.5.

After a distance d, D^′= D + 2dtanφ. This gives the following relation for V : V = D²

D^′2 = D²

(D + 4dntanφ)² (2.11)

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D D’

d f

Figure 2.5: The divergence of the laser beam after leaving the fiber. D is the diameter of the fiber core, D^′ is the diameter of the diverged laser beam, d is the distance between fiber and cantilever and φ is the angle of divergence.

To see the relation between the measured signal by the detector and the distance between fiber and cantilever we take the following laser/fiber constants.

D = 9µm, φ = 7^o, λ=1550nm and the reflectivity of the cantilever with an aluminium coating Rc=0.85. It/I0 is plotted versus d in figure 2.6. For large distances the signal is dominated by the reflection of the fiber. Most of the light reflected from the cantilever is lost due to the divergence of the laser beam.

As the cantilever moves closer to the fiber more and more light gets reflected back into the fiber and the visibility increases until at d = 20µm the cantilever reflection starts to dominate. At very small distances the signal approaches the reflection coefficient of the cantilever and almost all light is reflected back into the fiber.

I /I^T ⁰

20 40 60 80 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

d( m)m

2 4 6 8 10

0

d( m)m I /I^T ⁰

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 2.6: The calculated irradiance detected divided by the irradiance of the incoming laser of a cantilever fiber interference as a function of the distance, d, between the cantilever and the fiber.

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2.2 Forces

In the previous section the detection mechanism for the bending of the cantilever under influence of a tip sample force was discussed. The different types of forces that dominate in an AFM/MFM will be discussed in this section. As the tip-sample distance r changes, the forces working on the tip will also change.

For large tip-sample distances long range forces dominate such as magnetic or electrostatic forces. For small tip-sample distances the Van der Waals force dominates.

2.2.1 The Van der Waals force

The Lennard-Jones potential is shown in figure 2.7. The right attractive part of the curve is caused by the Van der Waals force. The left repulsive part is caused by the Pauli exclusion principle. For large tip-sample distances the Van der Waals force decreases to zero. As the distance decreases the attractive Van der Waals force increases. This increase continues until the atoms of the tip and sample come so close together that the electron clouds begin to repel each other due to the Pauli exclusion principle. Around a few angstrom the repulsive force due to the Pauli exclusion principle is just as large as the attractive Van der Waals force and no net force works on the cantilever. When the distance decreases even further the tip and sample are in contact. The slope of the Lennard-Jones potential is steep at low tip-sample distances. This means that the repulsive force grows fast when the tip is pushed further into the sample. If a stiff cantilever is used, or a soft surface, this force will deform the surface instead of bend the cantilever. It is therefore important to use the right cantilever for the right type of surface. This problem can be prevented by keeping the tip- sample distance larger. This can be done by operating the AFM in tapping mode, in the non contact region, which will be discussed in section 2.6.

Force

distance r repulsive

attractive contact region

non contact region

Figure 2.7: The force as a result of the Lennard-Jones potential is given by F = _r^C¹³−_r^D⁷ Here C and D depend on the tip sample properties.

The Van der Waals force is an intermolecular attractive force based on dipole interactions. There are three types of Van der Waals forces. The first type is

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the orientation interaction force between two molecules. Both molecules have an electric dipole moment and tend to align in order to reach energy minimalization.

The second force is between two molecules, one with and one without an electric dipole moment. The electric field induced by the dipole moment polarizes the other molecule. This other molecule gets polarized by the electric field of the first molecule and thus obtains a dipole moment. This dipole moment influences the first molecule again as well as other surrounding molecules. The dispersion interaction is the third type of Van der Waals force and is similar to the induced dipole interaction. The difference is that the initial dipole is not a permanent dipole but a dipole due to spontaneous fluctuations of the electric field. This interaction is the most important for AFM measurements because normally the tip and sample do not have any permanent dipoles. The dipole moment due to spontaneous fluctuations of an atom, p1creates an electric field, E1∝ p¹/r³, at a distance r. This electric field induces a polarization in a second atom at r, p2. If the polarizability of the second atom is α, p2∝ αE1. The energy of a dipole with dipole moment p placed in a field E is V = pE. So the potential of the dispersion interaction, and also for the induced dipole interaction, can be given by equation 2.12:

V = p2E1∝ αE1²∝ αp²₁

r⁶ (2.12)

Despite the different nature of the three Van der Waals forces they all have the same potential dependence on distance, ∝ r¹⁶. If the energy has a _r¹⁶ dependence the force, F ∝ ^∂V_∂r, has a _r¹7 dependence. This is the attractive, right, part of the Van der Waals curve. The left part of the curve was dominated by the Pauli exclusion principle. This part of the potential can be approximated by a _r¹12 dependence with an opposite sign as the Van der Waals potential. The Lenard-Jones potential is thus given by V (r) = _r^A¹²−_r^B⁶, where A and B depend on the tip sample properties.

2.2.2 The electrostatic force

If the tip and the sample are not grounded, charge can accumulate in one of them generating a voltage difference between the two. A capacitance C is created and the force between tip and sample can be given by equation 2.13 [6]:

F = −V² 2

∂C

∂r, (2.13)

where V is the potential difference between the tip and the sample. By con- necting the sample electrically to the cantilever the electrostatic force can be eliminated.

2.2.3 The magnetic force

In order to measure the magnetic force of a ferromagnetic sample, the tip has to be ferromagnetic. In most cases this is just a normal AFM tip with an extra ferromagnetic coating on the outside. It is complicated to derive the magnetization vector field of a tip and several simplifications are made in order to model this. The ’effective domain model’ [7] approximates the magnetization of the tip by a prolate spheroid with uniform magnetization as is shown in figure

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2.8. Outside this spheroid the magnetization is zero. The magnetization is also constant and cannot be changed by external magnetic fields, like the magnetic field produced by the sample. Although these simplifications describe the real tip well it is often difficult to assign the correct magnetic moments to the tip.

A further simplification is made in the ’point probe approximation’ [8]. The representation of such a tip can be seen in figure 2.8 (c).

(a) (b) ( ) c

Figure 2.8: Three schematic representations of magnetic tips. (a)A tip with a ferromagnetic coating. (b)An MFM tip as modeled by the effective domain model. (c) An MFM tip as modeled by the point probe approximation.

The magnetization of the tip in the point probe approximation is considered to be inside an infinitesimal point. A multipole expansion of the magnetic field of a tip results in a magnetic monopole and dipole moment as the first two most important terms. In general the magnetic monopole can be neglected and only a dipole moment remains. However there is a magnetic monopole moment that needs to be taken into account in the point probe approximation since the magnetic stray field produced by the sample decreases with distance according to its decay length. If the decay length is small only a part of the tip senses the sample as is shown in figure 2.9 (a), only the bottom part of the tip senses the magnetic field of the sample and this can be considered as a magnetic monopole. In figure 2.9 (b) the decay length of the sample is larger and the entire tip senses this field. In this case there is no contribution from the magnetic monopole moment.

(a) (b)

Figure 2.9: (a)An MFM tip in close proximity to a magnetic sample producing a magnetic field with a short decay length. (b) The same tip close to a sample with a long decay length.

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The infinitesimal point in the point probe approximation contains thus a magnetic monopole and a magnetic dipole and the force working on the tip can be given by equation 2.14 [9]:

F = µ0(q + m · ∇)H (2.14)

Here q and m are the effective magnetic monopole and dipole moment respec- tively and H is the magnetic field produced by the sample. In static mode the force is measured directly. In tapping mode, which will be discussed in paragraph 2.4, the derivative of the force with respect to the sample distance is measured [10].

∂F

∂z = µ0(∂q

∂z+ q∂H

∂z +∂m

∂z

∂H

∂z + m∂²H

∂z ) (2.15)

The magnetic dipole and monopole moment also depend on z, the height of the tip with respect to the sample, since for larger tip sample distances less of the tip senses the magnetic field of the probe. Even with all these simplifications it is still hard to give a good description of the tip. MFM is therefore often used as a qualitative measurement to show the magnetic contrast of the sample and not so much a quantitative method to measure the exact size of the stray field of the sample.

The assumption that the sample magnetization is not affected by the tip magnetic field and the other way around is only valid for hard magnetic tips and samples. A soft magnetic sample is a sample where the stray field is small.

However the field is also sensitive to external magnetic fields and imaging such a sample with a hard magnetic tip will distort the magnetic structure of the sample. A soft magnetic tip also has a small magnetization so the forces working here are much smaller. To still be able to image the magnetic structure the force constant of the cantilever is also smaller. As a general rule the magnetic anisotropy field, Hk the field at which the magnetic structure changes, should be larger than the external magnetization H [11]:

(Hk)^sample> (H)^tip, (2.16) and

(Hk)^tip> (H)^sample (2.17)

2.3 Cantilever dynamics in static mode

Most cantilevers are rectangular beams with a tip on the free end side. The other side is fixed to a much larger wafer as is shown in figure 2.10 (a).

The cantilever has a length l, a width w and a thickness t. Normally l ≫ w ≫ t. The tip is closest to the sample so it is assumed that all the forces that act on the cantilever, act on the tip. The cantilever can bend in the x-, y- and z-direction. If the bending in the x- and y-directions is small, the laser remains aligned at the back end of the cantilever and there is no change in the interference signal. Since l ≫ w ≫ t, the bending of the cantilever in the z-direction, the direction of the thickness dominates. Therefore we consider

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l

t w

ltip

cantilever

Si wafer

(a) (b)

( )c tip

Figure 2.10: Schematic view of the cantilever on a wafer. (a) The Si wafer with at the top the cantilever. (b) A zoomed in top view of the cantilever. (c) The side view of the cantilever.

only bending in the z-direction¹. Bending in the z-direction can happen due to three different forces, a force in the x-, y- or z-direction. If the bending of the cantilever is small, the cantilever can be approximated by a perfect spring. The relation between the bending and the force can then be described by Hook’s law [12]:

Fi= −ki∆z (2.18)

Here F is the force acting on the cantilever, k is the spring constant and ∆z is the deflection in the z-direction. The subscript i can be taken as x, y or z.

The cantilever bending resulting from a force in the z-direction is shown in figure 2.11. Two cross-sections are made and figure 2.12 (a) shows a beam element of length L between two cross-sections, (b) shows a cross-section. Here R is the cantilever curvature radius, L is the original length of the cantilever.

∆L is the extension of L at position z, where z is the distance from the neutral plane (the plane where ∆L=0). z can thus be seen as ∆R so ∆L/L = z/R since the relative increase in circumference equals the relative increase of the radius of a circle.

The Young’s modulus, E, of a material is a measure for the stiffness of the material and can be calculated by dividing the stress by the strain [13] and is given by equation 2.19:

E = F (y)L0

A0∆L (2.19)

F (y)² is the force at a distance y acting on a small strip of the cross-section, the area dS in figure 2.12, due to the bending caused by Fz. A0 is the cross- sectional area on which the force is applied. Because the force above the neutral line (z=0) points at the opposite direction as the force under the neutral line the forces cause a bending moment Mz.

1If the detection method used is not an interferometer but for example a quadrant detector, bending in the x- and y-direction can be detected. Nevertheless, with a quadrant detector the bending in the z-direction still dominates.

2Here F (y) is not the force in the z-direction caused by the tip-sample interaction.

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y z

x F

^z

Figure 2.11: Schematic view of a bended cantilever due to a force in the z- direction. The force is actually working on the tip but to make the image clearer it has been drawn at the top of the cantilever. The black cantilever is for Fz=0 and the red cantilever is bended under a force in the z-direction.

With the help of figure 2.12 and equation 2.19 an equation for the force working on A0can now be written as dF (y) = ^Ez∆S_R(y) and the bending moment is given by:

Mz(y) = Z

S

zdF (y) = Et³w

12R(y) (2.20)

Mz(y) is the bending moment at a point on the y axis due to Fz on y = l.

This can also be given by Mz(y) = Fz(l − y). If the bending of the cantilever is small the deflection of the cantilever at point y, u(y), can be given by 1/R(y) =

∂²u/∂y². To calculate the deflection at point y = l we have to solve [13]:

∂²u

∂y² = 12Mz

Et³w = 12Fz

Et³w(l − y) (2.21)

Since the cantilever is fixed at the wafer side and cannot move with respect to the dither the boundary conditions become: u|y=0 = 0 and ^∂u_∂y|y=0= 0, this gives:

u(l) = ∆z = kzFz = 4l³

Et³wFz (2.22)

for the force in the z-direction. The force in the y-direction, Fy, also results in a bending moment My as can be seen in figure 2.13. The cantilever deflects in the same way but reacts differently on the size of the force. Here My= Fyltip

with ltip as the length of the tip. The first part of equation 2.21 is still valid and Mz can be replaced by My which gives:

∂²u

∂y² = 12Fyltip

Et³w (2.23)

With the same boundary conditions as for the force in the z-direction this leads to the deflection resulting from a force in the y-direction.

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L L+DL z

R

t

w z

dS

(a) (b)

dS

A⁰ z

x z

x

z y z

Figure 2.12: (a) Beam element of length L between two cross-sections. The dashed line represents the cantilever in equilibrium while the normal line corresponds to the bended cantilever. (b) Front view of a cross-section of a bended cantilever. The entire rectangle is the cross-sectional area A0and dS is a small stripe of that area.

u(l) = ∆z =6Fyltipl²

Et³w = 3ltip

2l kzFy (2.24)

The bending in the z-direction due to Fxis much harder to calculate. There are two separate motions of the cantilever when a force in the x-direction is working as is depicted in figure 2.14. First of all a bending in the x-direction will occur. If the bending in the x-direction is small the measured signal does not change. In general this is the case and we will not discuss this motion. The cantilever will also make a torsional motion which is detected by the system.

The torsional motion is much harder to calculate than the previous cantilever motions and only the result will be given [14]:

∆z = 2l³_tip

l⁴ k²_zF_x² (2.25)

The forces in the three directions have been discussed and Hook’s law can be written as:

∆z =2l_tip³

l⁴ k_z²F_x²+3ltip

2l kzFy+ kzFz (2.26) Therefore the largest contribution to ∆z is from the force in the z-direction since the length of the cantilever is much longer than the length of the tip, l ≫ ltip. Typically the length of the cantilever is between 125 and 450 µm and the length of the tip is around 4 µm so from now on we will only consider forces acting on the tip in the z-direction.

2.4 Cantilever dynamics in oscillation mode

In section 2.3 the interaction of a static cantilever with a sample was discussed.

In static mode the cantilever bends under a force from the sample and this bending is measured. Because the force required to bend the cantilever is quite large you have to be in contact with the sample surface in order to make images.

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y z

x

F

^y

Figure 2.13: Schematic view of a bended cantilever due to a force Fy. The black cantilever is for Fy=0, the red cantilever is bended under a force in the y-direction.

z

x y y

z

Fx x

Fx

(a) (b)

Figure 2.14: Schematic view of a bended cantilever due to a force in the x- direction. The black cantilever is for Fx while the red cantilever is bended under a force in the x-direction. (a) The cantilever is bended in the x-direction.

(b) The cantilever makes a torsional motion due to a force in the x-direction.

This mode is known as contact mode. In most cases this is undesirable since the tip or sample can easily be damaged. Also in static mode it is impossible to image long range forces since these forces are typically small. In tapping mode, which is a non-contact mode, the tip taps the surface or even does not touch the surface at all.

The dither is a piezo element that moves under the influence of an electric field. The chip of the cantilever is attached to the dither so if the dither moves the cantilever moves as well. By applying an AC voltage to the dither the cantilever moves with the same frequency. By tuning the frequency close to the resonance frequency of the cantilever the amplitude of the cantilever increases.

By feeding the detector signal to a lock-in amplifier more information can be derived from the signal. The oscillation amplitude, phase and frequency of the tip can be determined. This method is more sensitive to tip-sample forces than static mode is. In this section the reaction of the cantilever to tip-sample forces

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will be discussed while the cantilever is oscillating. As was discussed in section 2.3 only a force in the z-direction is considered.

If the frequency of the dither is close to the resonance frequency of the cantilever the amplitude of the end of the cantilever will be much larger than that of the dither. The increased value of the amplitude at the resonance frequency is determined by the quality factor Q. Q is the total energy stored divided by the total energy lost during an oscillation cycle and is defined as _∆ω^ω⁰ as can be seen in figure 2.15.

w (a.u.)

Amplitude(normalized)

1

1/ 2

Dw

w

Q=w/Dw

Figure 2.15: Resonance peak of a cantilever. Q is defined as the resonance frequency divided by the width of the peak when the amplitude is √¹

2 of the maximum amplitude.

The equation of motion for the oscillating cantilever in the presence of an external force, F0, is given by equation 2.27:

m∂²z

∂t² + γ∂z

∂t + kz = F0, (2.27)

where m is the mass of the cantilever, k is the spring constant of the cantilever and γ is the dissipation term given by γ=mw0/Q. The eigenfrequency ω0 is related to the spring constant by: ω0=p

k/m. z is the position of the cantilever free end with z = 0 as the equilibrium position of the cantilever free end when there is no tip-sample interaction. If we divide the equation by m and define δ=γ/m and A=F0/m we get:

∂²z

∂t² + δ∂z

∂t + ω²₀z = A (2.28)

The cantilever is driven with a harmonic oscillation A=A0cos(ωt). Without

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tip-sample interaction and without damping the resonance frequency is ω0. ωris the resonant drive frequency, this is the resonance frequency with damping taken into account but still without tip-sample contact. Both resonance frequencies can be calculated:

ω0= rk

m (2.29)

ωr= ω0

r 1 − 1

Q² (2.30)

The phase difference between the drive and tip oscillation is ϕ. When the cantilever is driven at ωr without tip-sample interaction the phase difference between the tip and the drive oscillation is π/2. This is the general result for a spring with one end fixed and one end free. The additional 1/Q²factor for the resonant drive frequency is due to the damping that causes a difference between the natural resonance frequency of the cantilever and the resonant drive frequency. If the dissipation term is small, Q will be large and there will be little difference between the natural resonance frequency and the resonant drive frequency. In the case that there is also tip-sample interaction an additional force acts on the cantilever. It is important to know how this extra force changes the resonance frequency, the resonant drive frequency, the phase and the amplitude of the oscillation of the cantilever. The result of this force can be seen in figure 2.16, where the amplitude and phase are plotted versus drive frequency for a cantilever with and without tip sample interaction.

w (a.u.)

Z(a.u.)0

Dw^r

j p/2

-p/2

w (a.u.) Dwr

(a) (b)

Dz⁰

Figure 2.16: (a) Amplitude versus drive frequency of a cantilever, Z0 is the amplitude of the free end of the cantilever. (b) Phase versus drive frequency of a cantilever. The black line is for a cantilever without tip sample interaction and the red line for a cantilever with an repulsive tip-sample force.

This tip-sample force is given by Fts and corresponds with A, Ats is defined as Ats= Fts/m. This gives the new equation of motion in formula 2.31

∂²z

∂t² + δ∂z

∂t + ω²₀z = A0cos ωt + Ats(z) (2.31)

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Ats changes the equilibrium position about which the cantilever oscillates. For small z a Taylor expansion of Ats with z0as the equilibrium position gives:

Ats(z) = Ats(z0) +∂Ats

∂z (z0)(z(t) − z0) + O(z(t) − z0)² (2.32) Neglecting the higher order terms for small tip-sample interactions and defining e

z = (z(t) − z0) we get:

∂²ez

∂t² + δ∂ez

∂t + (ω²₀+∂Ats

∂z )ez = A0cos ωt (2.33) With kef f =q

k − ^∂F_∂z^ts as the new effective spring constant and fω0=p

(kef f/m) = q

ω₀²+^∂A_∂z^ts as the new resonance frequency. A general solution to 2.33 is given by equation 2.34:

e

z(t) = ezs(t) + Z0cos(ωt + ϕ) (2.34) Here ezs(t) is the stationary solution. With δ > 0 free oscillations are damped and ezs(t) approaches 0 for large t. Therefore the movement of the tip of the cantilever is completely independent of the starting conditions for large t. The new amplitude and ϕ can be now be calculated and are give by:

Zf0= A0

r

(fω₀²− ω²)²+^ωe²0ω² Q²

(2.35)

and

tan ϕ = ωf0ω

Q(ω²− fω₀²) (2.36)

The new resonant drive frequency, fωr, can now be calculated from equation 2.30.

f ωr= ω0

s

1 − ∂Fts

k∂z − 1 2Q² =

s

ω²_r−ω₀²^∂F_∂z^ts

k (2.37)

The resonant drive frequency is determined by three terms. The first is the original resonance frequency, the second term compensates for the tip sample force and the third term compensates for the dissipation.

The change in resonant drive frequency is given as:

∆ωr= fωr− ωr= ωr( s

1 − ω₀² kω_r²

∂Fts

∂z − 1) (2.38)

For |^ω

2 0

kω_r²

∂Fts

∂z | ≤ 1 a Taylor expansion can be made and the change in resonant drive frequency can be given by ∆ωr≈ −^ω_2k∂z⁰^∂F^ts .

If there is a tip-sample interaction the new phase difference can be calculated with equation 2.36 and gives:

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Table 2.1: Overview of the changes in amplitude, resonance frequency and phase due to a tip-sample interaction.

without tip-sample interaction with tip-sample interaction approximated difference Z0= q ^A⁰

(ω²₀−ω²a)²+^ω

2 0ω2

a Q2

A0

q

(ωe²0−ω²a)²+^ω0e²^ω²a Q2

−^2Z₃^max^√_3k^Q^∂F_∂z^ts

ωr ωr

q

1 −_kω^ω⁰²²_r^∂F_∂z^ts ωr(q

1 −_kω^ω⁰²²_r^∂F_∂z^ts − 1) ϕ = arctan(_Q(ω^ω2⁰^ω

−ω0²)) = ^π₂ arctan( ^ωe⁰^ω

Q(ω²−ωe0²)) −^Q_k^∂F_∂z^ts

e

ϕ(ω0) = arctan( k

Q∂Fts

∂z

) (2.39)

For|Q∂Fts^k

∂z | ≥ 1 a Taylor expansion can be made and the phase difference is given by the next equation:

∆ϕ = eϕ(ω0) − ϕ(ω⁰) ≈ −Q∂Fts

k∂z (2.40)

Now we take a look at the amplitude difference between an oscillation with and without tip sample interaction. The maximum slope of the amplitude versus frequency curve is at [15]:

ωa= ωr(1 ± 1

√8Q) = ω0

r

1 − ∂Fts

k∂z(1 ± 1

√8Q) (2.41)

For small changes in frequency the slope can be taken linearly and the amplitude difference can be given by equation 2.42.

∆Z0≈ ∆ωr

∂ fZ0

∂ωr

(ωa) ≈ −(2 eZmaxQ 3√

3k )∂Fts

∂z (2.42)

To summarize, the parameters measured by the lock-in-amplifier, which are the resonance frequency, the amplitude and the phase of the cantilever all change under a tip-sample force, which is shown in table 2.1.

2.5 Sensitivity

In order to image the smallest features of a sample the noise level should be minimized; as this is limiting the resolution. The minimal detectable force is obtained when the signal change due to tip-sample interaction is as much as the highest noise term. Besides noise coming from the environment, such as building vibrations and acoustical noise, there are three different types of noise. The first noise type is the noise in the displacement sensor. The second noise type is the noise generated in the oscillation control amplifier and other electronics. And the third noise term is the thermal noise that causes vibrations

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of the cantilever. The first two types of noise are very small when an optical interferometer technique is used to measure the displacement of the cantilever and the main contribution to the noise is from the thermal noise [15].

To be able to measure the change in force between the tip and the sample, this force should give an amplitude difference larger than the noise in the amplitude. The change in amplitude due to thermal noise is given by 2.43 [15].

N = Z0(T ) Z0(0) =

r4kbT QB kωa

, (2.43)

where kbis Boltzmann’s constant, T is the temperature and B is the bandwidth.

The latter is a measure for the measuring time. ∆A should be at least as large as N , so the minimal detectable force difference can be given by [15]:

(∂Fts

∂z )min= 1 A0

s

27kkbT B

ω0Q (2.44)

A high value of Q thus gives a greater sensitivity. Q can be increased by operating the system in vacuum to reduce the damping due to air, or by cooling down to change the force constant of the cantilever. However, increasing Q will restrict the bandwidth of the system and might lead to too long measuring times since B and Q are related through B = _2Q^ω⁰ ³. A way to solve this problem is to use a frequency modulation mode to detect the force acting on the cantilever.

How the frequency modulation modes work will be discussed in the next sections but for now this mode can be considered as a mode where the frequency is always at resonance and the difference between ωr and fωr is a measure for the force acting on the tip.

The general solution for the equation of motion was given by equation 2.34.

So far we have ignored the stationary term in the solution because for large t it can be neglected. However if we look at what happens to the amplitude of the oscillation of the tip after the force on the tip changes, we need to take into account how long it takes for the amplitude to settle down to its final value.

The stationary solution for ω0> δ, which is almost always valid, is given by:

e

zs(t) = e⁻^δt² cos(ωt + eϕ) = e⁻e^ω0t₂

Q cos(ωt + eϕ) (2.45) As can be seen from this solution a large Q prevents the amplitude to change quickly if there is a change in tip and sample force. However, for high sensitivity Q needs to be large. There is thus the choice between a high sensitivity or a fast response time. In vacuum conditions at low temperatures in general Q will be large. If vacuum is needed it might not be possible to get a low enough Q to still be able to have a fast enough response time. Amplitude modulation will not work in those cases and frequency modulation is a good alternative. Since the frequency changes really fast, within one oscillation period, the bandwidth of the system is no longer dependent on Q. This means that Q can be taken high to optimize sensitivity. The minimal detectable force difference was calculated and only the result is given here [16]:

3This bandwidth is determined by the measuring time. It is different than the bandwidth used in the definition of Q. When talking about AFM sensitivity this new bandwidth is always used instead of the old one. This new value can also be seen in equation 2.45

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(∂Fts

∂z )min=2k∆ω0

ω0

= 1 A0

s

4kkbT B

ω0Q (2.46)

This is already better than for slope detection but the most important improve- ment is that B and Q are not related anymore. The amplitude still varies slow but since you now measure the frequency there is no need to wait for the amplitude to reach its new value. Q is determined by the cantilever and its surrounding and B is now completely determined by the frequency modulation electronics.

2.6 Operating modes

2.6.1 Amplitude Modulation

As was discussed in the previous section a good alternative for static mode is the lock-in mode, an amplitude modulation mode. In amplitude modulation mode the difference in amplitude is used as a measure for the force between tip and sample. The amplitude is kept constant by adjusting the z-scanner during scanning in the x- and y-direction. A schematic view of this mode can be seen in figure 2.17. The red color indicates a phase shift with respect to the detected phase, which is in this case 90 degrees at resonance. The dither is driven with a fixed amplitude and frequency. The detector signal is AC coupled into a gain to get a pure AC signal with the desired amplitude. The analog signal is then converted to a digital signal that enters the PLL unit. In lock-in mode the PLL unit is only used as a lock-in amplifier to extract the amplitude and the phase.

These signals can be used for the feedback loop to move the z-piezo. The raw signal is also filtered and can be monitored. Under ambient conditions this is the most used mode to operate an AFM.

normal force

gain

1-512 ADC Digital

PLL

Output Gain

1 0.1 0.01 0.001

Drive Amplitude AC

Coupled

Detector Output

Select

SPM 100 Filter PC

Probe excitation

Probe drive

z

x,y,z

Phase Amplitude

Figure 2.17: Schematic view of the lock in mode. The red color indicates a phase shift with respect to the detected phase.

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2.6.2 Frequency Modulation

If the cantilever operates at low temperatures and at low pressures the quality factor will increase. This increase in Q gives a higher sensitivity but also has two drawbacks. First of all the amplitude used as a measure for the tip-sample force needs a long measuring time to become stationary, as was shown in the previous paragraph. This leads to a low scanning speed in amplitude modulation mode.

Another disadvantage is that the resonance peak becomes very narrow. If the cantilever is now driven with a fixed frequency and the resonance frequency changes due to tip-sample interactions the peak will shift and the cantilever is driven off resonance and imaging becomes impossible. The best way to solve this is to add an additional feedback loop to the system to constantly change the drive frequency in order to match the resonance frequency. This is the principle of the two frequency modulation modes.

Self Oscillation Mode

In self oscillation mode the detector signal is phase shifted and used to drive the dither. The variable loop gain is used to make sure the drive amplitude remains the same. In resonance the phase difference between the dither and the tip of the cantilever is 90 degrees. The detected signal of the cantilever is used to drive the dither directly. Therefore a phase shift is needed in order to match the new and the old drive oscillation in a continuous fashion, see figure 2.18. On average the frequency of the tip is equal to the frequency of the dither but on short time scales the frequency changes a bit to adjust to the new resonance frequency. The new resonance frequency will get amplified and then gets fed back into the system as the new drive signal. This means that the drive frequency shifts together with the resonance frequency and the cantilever always oscillates in resonance. In this way it is possible to image with high Q. A constant height distance between your tip and sample can be achieved if

∆f , the frequency shift, is used for z-feedback. There are two disadvantages in this mode. If two resonance peaks of the cantilever are relatively close together the drive frequency can jump between these peaks. A second disadvantage is that the movement of the tip is not completely sinusoidal. This means that the drive amplitude is also not completely sinusoidal and this can enhance itself and distort the signal.

Phase Locked Loop Mode

The phase locked loop mode (PLL mode) solves the problems of the self oscillation mode. In figure 2.19 the schematic view of the PLL loop can be seen. In the PLL mode the drive oscillation is determined by an external oscillator. This gives a perfect sinusoidal drive oscillation. The signal of the tip oscillation is then analyzed to give the frequency and the phase. The sinusoidal drive signal is forced to oscillate with the measured frequency and the phase is now shifted by a predetermined amount to force the cantilever to oscillate at the right resonance peak. As for self oscillation mode the feedback to the z-piezo is governed by the change in resonance frequency. In figure 2.20 the narrow resonance peak can be seen. As the tip-sample force increases the peak shifts. In AM mode the peak was broader and the amplitude difference was used to measure. This can not be done with the narrow peak since there would be no amplitude left

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normal force

gain

1-512 ADC Digital

PLL

Output Gain

1 0.1 0.01 0.001 AC Coupled

Detector Output

Select

SPM 100 Filter PC

Probe excitation

Probe drive

z

x,y,z

Df Amplitude

Phase

Phase shift

Variable loop gain

Figure 2.18: Schematic view of the self oscillation mode. The red color indicates a phase shift with respect to the detected phase.

even after a small frequency shift. Instead the PLL mode shifts the phase of the detected signal to make sure that the drive signal is on the new resonance frequency. The cantilever is thus always driven on resonance and the amplitude of the detected signal is thus almost constant for different tip-sample forces.

The NCO, numerically-controlled oscillator, is used to make the detected signal phase continuous. The phase shift is determined by doing a resonance sweep and checking what phase corresponds to the maximum amplitude. And because the phase shift is always the same in resonance the cantilever always oscillates at the right resonance peak. This is the most stable mode for high Q factors but it is also a bit slower because the signal has to be analyzed before a change in drive signal can be made.

normal force

gain

1-512 ADC Digital

PLL

NCO

Digital Phas shift DAC

Output Gain

1 0.1 0.01 0.001

Drive Amplitude AC

Coupled

Detector Output

Select

SPM 100 Filter PC

Probe excitation

Probe drive

z

x,y,z

Df Amplitude

Phase

Figure 2.19: Schematic view of the PLL loop. The red color indicates a phase shift with respect to the detected phase.

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Dw

Amplitude(a.u.) Phase(a.u.)

Frequency (a.u.)

Figure 2.20: Amplitude and phase shift as a result of the repulsive tip-sample interaction. The PLL loop forces the drive signal to drive the cantilever near resonance with a phase of 90^o. As the tip-sample interaction changes the resonance frequency changes and thus the phase changes. By keeping the phase locked at 90^o the cantilever always oscillates on resonance.

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Chapter 3

Experimental Setup and Problems

3.1 The AFM

The AFM that was used in this thesis is the AttoAFM-I. This AFM is designed by Attocube to work at low temperatures. The setup can be divided in four different elements: the scan head, the optics, the extension stick and the electronics. The scan head as well as the extension stick are made of non-magnetic materials allowing operation in an externally applied magnetic field at cryogenic temperatures.

3.2 Piezoelectrics

Because an important part of the AFM scan head consists of piezoelectric elements, the basics of piezoelectrics will be discussed first. Piezoelectric materials become polarized when they are subjected to a mechanical force, resulting in a voltage proportional to the applied force. On the other hand an electric field lengthens or shortens the piezoelectric material depending on the polarization of the field. Only small length differences can be obtained (even with large volt- ages) which is good for the actuation application such as AFM scanners where only small surface areas need to be scanned at a time.

The piezo elements used here are PZT piezos and consist of a lead zirconate titanate ceramic. Piezoelectric ceramics have a perovskite crystal structure.

Above a critical temperature, the Curie temperature, the perovskite crystals have a cubic symmetry and no dipole moment as is shown in figure 3.1a. If the temperature is below the Curie temperature the crystal structure is slightly distorted and gets a tetragonal symmetry as is shown in figure 3.1b. The central T i⁴⁺ or Zr⁴⁺ is closer to the top than to the bottom of the unit cell creating a dipole moment. These dipole moments tend to align and form domains. The direction of polarization of the different domains is random so the net polarization of the piezo element is zero. By exposing the element to a direct electric field the domains that are aligned with the electric field increase and the domains that are not aligned decrease. This means the piezo element lengthens in the direc-

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Pb

Ti /Zr O

4+

2+

2- 4+

(a) (b)

Figure 3.1: a) A perovskite crystal with the temperature above the Curie temperature having a cubic crystal structure. b) A perovskite crystal with a temperature that is lower than the Curie temperature. The crystal structure is tetragonal and has an electric dipole moment.

tion of the applied electric field. When the electric field is removed the domains remain aligned and the piezo now has a permanent polarization and elongation.

Ferromagnetic materials show hysteresis and the same behavior can be seen for ferroelectric materials. In figure 3.2 the polarization can be seen as a function of the electric field. In figure 3.3 the strain, or the relative increase(decrease) in size can be seen as a function of the electric field. The relative increase in the direction of the applied electric field is accompanied by a corresponding but smaller decrease in the direction perpendicular to the applied field. This is no problem for the used actuators because they can only move the piezostack in the direction of the applied field. In the other direction the piezo is fixed with only one end while the other end is free so deformation in this direction will not move the piezo stack.

A piezo element can depolarize and loose its piezoelectric properties. This occurs when the temperature approaches, or is higher than, the Curie temperature, or when a large voltage creating an opposite electric field is applied or due to a large force that is applied to the piezo. The piezo thus has to be treated with care to avoid damage. If the temperature is decreased the hysteresis graph will change. The maximum polarization will remain the same but when the electric field is decreased there is less thermal motion and the domains tend to remain intact longer. Therefore at zero electric field the polarization is higher.

A change in electric field now responds in a smaller change of the polarization. This means that the scan area at low temperatures will get smaller for the same applied voltage. The range over which the sample can move is very small. Moreover when the system is prepared for measurements the sample can move a bit with respect to the tip and a tip crash can easily occur. To get to the interesting areas of the sample there are three additional piezo elements, the positioners, that provide a much larger movement of the sample. So the scanners for the x-, y- and z-direction are used for small continuous movements whereas the positioners are used to make discrete larger steps.

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P(a.u.)

E(a.u.)

Figure 3.2: Hysteresis curve of the polarization P versus the electric field E. The arrows indicate how the polarization changes when the electric field changes.

The principle of operation of the positioners is seen in figure 3.4. The piezo element is fixed on one side to the piezostack and on the other side to a mass m2, the inertial weight. There is also a main body, m1, that tightly surrounds m2. If the piezo tube is slowly extended the inertial force is smaller than the friction force between m2 and the main body so the main body will follow the movement of the piezo. This is shown in figure 3.4(a) and (b). If afterwards a fast voltage drop is applied to the piezo the inertial force is larger than the friction force and the piezo will shorten but the main body will stay behind.

This can be seen in figure 3.4(c).

The displacement of the main body after one cycle is called ∆x. ∆x depends on the extension of the piezo and on the masses m1 and m2. For the piezo to follow an ideal sawtooth voltage the piezo has to be able to retract very fast, this means that m2cannot be large. However m1has to be large in comparison to m2 in order to overcome the frictional force. ∆x can be expressed by the following equation:

∆x = m1

m1+ m2d (3.1)

here d is the extension of the piezo from figure 3.4 a to b. This can be repeated several times so the main body will move a large distance while the piezo element only has a limited range. Now the sample can move in three directions over quite large distances(∼6 mm) while the scan range is still small and accurate.

Eindhoven University of Technology MASTER The development of a low temperature MFM Meertens, Arjan