Bhushan B. Nanotribology and Nanomechanics: An Introduction

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212 Markus Morgenstern et al.

relax downward [134]. This corresponds to the situation in Fig. 5.19a, where the tip

is relatively far away and an inward relaxation of the two arsenic atoms is observed.

The considerablylarger attractive force in Fig. 5.19b, however, pulls the two arsenic

atoms toward the tip. All other arsenic atoms are also pulled, but they are less dis-

placed, becausethey have three bonds to the bulk,while the two arsenic atoms in the

neighborhood of an indium vacancy have only two bonds. This direct experimental

proof of the presence of tip-induced relaxations is also relevant for STM measure-

ments, because the tip–sample distances are similar during atomic-resolution imag-

ing. Moreover, the result demonstrates that FM-AFM can probe elastic properties

on an atomic level.

Imaging of Weakly Interacting van der Waals Surfaces

For weakly interacting van der Waals surfaces, much smaller atomic corrugation

amplitudes are expected compared to strongly interacting surfaces of semiconduc-

tors. A typical example is graphite, a layered material, where the carbon atoms are

covalently bonded and arranged in a honeycomb structure within the (0001) plane.

Individual graphene layers stick together by van der Waals forces. Due to the ABA

stacking, three distinctive sites exist on the (0001) surface: carbon atoms with (A-

type)andwithout(B-type)neighborin thenextgraphitelayerand the hollowsite (H-

site) in the hexagon center. In static contact force microscopy as well as in STM the

contrast exhibits usually a trigonal symmetry with a periodicity of 246pm, where

A-andB-site carbon atoms could not be distinguished. However, in high-resolution

FM-AFM images acquired at low temperatures, a large maximum and two diﬀerent

minima have been resolved, as demonstrated by the proﬁles along the three equiv-

alent [1-100] directions in Fig. 5.20a. A simulation using the Lennard–Jones po-

tential, given by the short-range interatomic van der Waals force, reproduced these

three features very well (dotted line). Therefore, the large maximum could be as-

signed to the H-site, while the two diﬀerent minima represent A-andB-type carbon

atoms [132].

Compared to graphite, the carbon atoms in a single-walled carbon nanotube

(SWNT), which consists of a single rolled up graphene layer, are indistinguish-

able. For the ﬁrst time Ashino et al. [133] successfully imaged the curved surface

of a SWNT with atomic resolution. Note that for geometric reasons, atomic resolu-

tion is only achieved on the top (see Fig. 5.20b). Indeed, as shown in Fig. 5.20b, all

proﬁlesbetween two hollowsites across twoneighboring carbonatoms are symmet-

ric [135]. Particularly, curve1 and 2 exhibittwo minima of equal depth, as predicted

by theory (cf., dotted line). The assumption used in the simulation (dotted lines in

the proﬁles of Fig. 5.20) that interatomic van der Waals forces are responsible for

the atomic-scale contrast has been supported by a quantitative evaluation of force

spectroscopy data obtained on SWNTs [133].

Interestingly, the image contrast on graphite and SWNTs is inverted with re-

spect to the arrangement of the atoms, i.e., the minima correspond to the position

of the carbon atoms. This can be related to the small carbon–carbon distance of

only 142pm, which is in fact the smallest interatomic distance that has been re-

5 Low-Temperature Scanning Probe Microscopy 213

426 pm

900 pm

Corrugation a.u.

Xe Xe

Xe Xe Xe

Corrugation a.u.

Fig. 5.20. FM-AFM images

of graphite(0001) (a) a single-

walled carbon nanotube

(SWNT) (b) and Xe(111) (c)

recorded at 22 K. On the right

side, line sections taken from

the experimental data (solid

lines) are compared to simu-

lations (dotted lines). A-and

B-type carbon atoms, as well

as the hollow site (H-site) on

graphite can be distinguished,

but are imaged with inverted

contrast, i.e., the carbon sites

are displayed as minima.

Such an inversion does not

occur on Xe(111)

solved with FM-AFM so far. The van der Waals radius of the front tip atom, (e.g.,

210 pm for silicon) has a radius that is signiﬁcantly larger than the intercarbon dis-

tance. Therefore, next-nearest-neighborinteractions become important and result in

a contrast inversion [135].

While experiments on graphite and SWNTs basically take advantage of the in-

creased stability and signal-to-noise ratio at low temperatures, solid xenon (melting

temperature T

= 161K) can only be observed at suﬃcient low temperatures [8]. In

214 Markus Morgenstern et al.

addition,xenonis a pure vander Waals crystaland, since it is an insulator, FM-AFM

is the only real-space method available today that allows the study of solid xenon

on the atomic scale.

Allers et al. [8] adsorbed a well-ordered xenon ﬁlm on cold graphite(0001)

(T < 55K) and studied it subsequently at 22 K by FM-AFM (see Fig. 5.20c). The

sixfold symmetry and the distance between the protrusions corresponds well with

the nearest-neighbordistance in the close-packed (111) plane of bulk xenon, which

crystallizes in a face-centered cubic structure. A comparison between experiment

and simulation conﬁrmed that the protrusions correspond to the position of the

xenon atoms [132]. However, the simulated corrugation amplitudes do not ﬁt as

well as for graphite (see sections in Fig. 5.20c). A possible reasonis that tip-induced

relaxations, which were not considered in the simulations, are more important for

this pure van der Waals crystal xenon than they are for graphite, because in-plane

graphite exhibits strong covalent bonds. Nevertheless, the results demonstrated for

the ﬁrst time that a weakly bonded van der Waals crystal could be imaged nonde-

structively on the atomic scale. Note that on Xe(111) no contrast inversion exists,

presumably because the separation between Xe sites is about 450pm, i.e., twice as

large as the van der Waals radius of a silicon atom at the tip end.

Atomic Resolution Using Small Oscillation Amplitudes

All the examplesabove described used spring constants and amplitudeson the order

of 40 N/m and 10nm, respectively, to obtain atomic resolution. However, Giessibl

et al. [137] pointed out that the optimal amplitude should be on the order of the

characteristic decay length λ of the relevant tip–sample interaction. For short-range

interactions, which are responsible for the atomic-scale contrast, λ is on the order of

0.1nm. On the other hand, stable imaging without a jump-to-contact is only possi-

ble as long as the restoring force c

A at the lower turnaround point of each cycle is

larger than the maximal attractive tip–sample force. Therefore, reducing the desired

amplitude by a factor of 100 requires a 100 times larger spring constant. Indeed,

Hembacher et al. [136] could demonstrate atomic resolution with small amplitudes

(about 0.25nm) and large spring constants (about 1800N/m) utilizing a qPlus sen-

sor [138]. Figure 5.21 shows a constant-height image of graphite recorded at 4.9K

within the repulsive regime. Note that compared to Fig. 5.20a,b the contrast is in-

verted,i.e., the carbon atoms appearas maxima.This is expected,because the imag-

ing interaction is switched from attractive to repulsive regime [131,135].

5.5.2 Force Spectroscopy

A wealth of information about the nature of the tip–sample interaction can be ob-

tained by measuring its distance dependence. This is usually done by recording

the measured quantity (deﬂection, frequency shift, amplitude change, phase shift)

and applying an appropriate voltage ramp to the z-electrode of the scanner piezo,

while the z-feedback is switched oﬀ. According to (5.2), low temperatures and high

Q-factors (vacuum) considerably increase the force resolution. In the static mode,

5 Low-Temperature Scanning Probe Microscopy 215

1.49

0.86

Fig. 5.21. Constant-

height FM-AFM image of

graphite(0001) recorded at

4.9K using asmall am-

plitude (A = 0.25 nm) and

a large spring constant

= 1800 N/m). As in

Fig. 5.20a, A-andB-site

carbon atoms can be dis-

tinguished. However, they

appear as maxima, because

imaging has been performed

in the repulsive regime

(courtesy of F. J. Giessibl;

cf. [136])

long-rangeforces and contact forces can be examined. Force measurements at small

tip–sample distances are inhibited by the jump-to-contact phenomenon: If the force

gradient ∂F

/∂z becomes larger than the spring constant c

, the cantilever cannot

resist the attractive tip–sample forces and the tip snaps onto the surface. Suﬃciently

large spring constants prevent this eﬀect, but reduce the force resolution. In the dy-

namic modes, the jump-to-contact can be avoided due to the additional restoring

force (c

A) at the lower turnaround point. The highest sensitivity can be achieved

in vacuum by using the FM technique, i.e., by recording Δ f(z)-curves. An alterna-

tive FM spectroscopy method, the recording of Δ f(A)-curves, has been suggested

by Hölscher et al. [139]. Note that, if the amplitude is much larger than the charac-

teristic decay length of the tip–sample force, the frequency shift cannot simply be

convertedinto force gradientsby using ∂F

/∂z= 2c

·Δ f/ f

[140].Severalmethods

have been published to convert Δ f(z) data into the tip–sample potential V

(z)and

tip–sample force F

(z) (see, e.g., [141–144]).

Measurement of Interatomic Forces at Speciﬁc Atomic Sites

FM force spectroscopy has been successfully used to measure and determine quan-

titatively the short-range chemical force between the foremost tip atom and speciﬁc

surface atoms [109, 145, 146]. Figure 5.22 displays an example for the quantita-

tive determination of the short-range force. Figure 5.22a shows two Δ f(z)-curves

measured with a silicon tip above a corner hole and above an adatom. Their po-

sition is indicated by arrows in the inset, which displays the atomically resolved

Si(111)-(7×7) surface. The two curves diﬀer from each other only for small tip0-

sample distances, because the long-range forces do not contribute to the atomic-

scale contrast. The low, thermally induced lateral drift and the high stability at low

216 Markus Morgenstern et al.

Δf (Hz)

z-displacement (nm)

2.0

–20

–40

–60

–80

0.5 1.0 1.5

Force (nN)

z-displacement (nm)

1.2

–1

–2

–3

0.4 0.6 0.8 1.0

3 nm

Short-range

force F

Total

force F

tot

Fig. 5.22. FM force spec-

troscopy on speciﬁc atomic

sites at 7.2K. In (a), an

FM-SFM image of the

Si(111)-(7×7) surface is

displayed together with two

Δ f(z)-curves, which have

been recorded at the posi-

tions indicated by the arrows,

i.e., above the corner hole

(brown) and above an adatom

(black). In (b), the total force

above an adatom (brown line)

has been recovered from the

Δ f(z)-curve. After subtrac-

tion of the long-range part,

the short-range force can be

determined (black line) (cour-

tesy of H.J. Hug; cf. [145])

0.4

0.3

0.2

0.1

0.2 0.4 0.6 0.8 1

–0.2

–1.0

Fig. 5.23. Principle of the 3-D force ﬁeld spectroscopy method (a) and a 2-D cut through

the 3-D force ﬁeld F

(x,y,z) recorded at 14 K (b). At all 32×32 image points of the 1nm

× 1 nm scan area on NiO(001), a Δ f(z)-curve has been recorded. The Δ f(x, y,z) data set

obtained is then converted into the 3-D tip–sample force ﬁeld F

(x, y, z). The shaded slice

(x, y = const, z)in(a) corresponds to a cut along the [100]-direction and demonstrates

that atomic resolution has been obtained, because the distance between the protrusions

corresponds well to the lattice constant of nickel oxide

5 Low-Temperature Scanning Probe Microscopy 217

temperatures were required to precisely address the two speciﬁc sites. To extract the

short-range force, the long-range van der Waals and/or electrostatic forces can be

subtracted from the total force. The grey curve in Fig. 5.22b has been reconstructed

from the Δ f(z)-curve recorded above an adatom and represents the total force. After

removing the long-range contribution from the data, the much steeper black line is

obtained, which corresponds to the short-range force between the adatom and the

atom at the tip apex. The measuredmaximum attractive force(−2.1nN) agrees well

with ﬁrst-principles calculations (−2.25nN).

Three-Dimensional Force Field Spectroscopy

Further progress with the FM technique has been made by Hölscher et al. [147].

They acquired a complete 3-D force ﬁeld on NiO(001) with atomic resolution (3-D

force ﬁeld spectroscopy). In Fig. 5.23a, the atomically resolved FM-AFM image of

NiO(001) is shown together with the coordinate system used and the tip to illustrate

the measurement principle. NiO(001) crystallizes in the rock-salt structure. The dis-

tance between the protrusions corresponds to the lattice constant of 417pm, i.e.,

only one type of atom (most likely the oxygen)is imaged as a protrusion. In an area

of 1nm×1nm,32×32 individual Δ f(z)-curves have been recorded at every (x, y)

image point and converted into F

(z)-curves.The Δ f(x, y,z) data set is thereby con-

verted into the 3-D force ﬁeld F

(x, y, z). Figure 5.23b, where a speciﬁc x–z-plane

is displayed, demonstrates that atomic resolution is achieved. It represents a 2-D

cut F

(x, y = const,z) along the [100]-direction (corresponding to the shaded slice

marked in Fig. 5.23a). Since a large number of curves have been recorded, Langkat

et al. [146] could evaluate the whole data set by standard statistical means to extract

the long- and short-range forces. A possible future application of 3-D force ﬁeld

spectroscopy could be to map the short-range forces of complex molecules with

functionalized tips in order to resolve locally their chemical reactivity. A ﬁrst step

in this direction has been accomplished on SWNTs. Its structural unit, a hexagonal

carbon ring, is common to all aromatic molecules.Like the constant frequency-shift

image of an SWNT shown in Fig. 5.20b the force map shows clear diﬀerences be-

tween hollow sites and carbon sites [133]. Analyzing site-speciﬁc individual force

curves extracted from the 3-D data revealed a maximum attractive force of about

−0.106nN above H-sites and about −0.075nN above carbon sites. Since the at-

traction is one order of magnitude weaker than on Si(111)-(7×7) (cf., Fig. 5.22b),

it has been inferred that the short-range interatomic van der Waals force and not

a chemical force is responsible for atomic-scale contrast formation on such nonre-

active surfaces.

Noncontact Friction

Another approach to achieve small tip–sample distances in combination with high

force sensitivity is to use soft springs in a perpendicular conﬁguration. The much

higher cantilever stiﬀness along the cantilever axis prevents the jump-to-contact,

but the lateral resolution is limited by the magnitude of the oscillation amplitude.

However, with such a set-up at low temperatures, Stipe et al. [148] measured the

218 Markus Morgenstern et al.

a) b) c)

Fig. 5.24. Consecutively recorded FM-AFM images showing the tip-induced manipulation

of a Ge adatom on Ge(111)-c(2×8) at 80 K. Scanning was performed from bottom to top

(courtesy of N. Oyabu; cf. [150])

distance dependence of the very small force due to noncontact friction between

the tip and sample in vacuum. The eﬀect was attributed to electric charges, which

are moved parallel to the surface by the oscillating tip. Since the topography was

not recorded in situ, the inﬂuence of contaminants or surface steps remained un-

known.

5.5.3 Atomic Manipulation

Nowadays, atomic-scale manipulation is routinely performed using an STM tip (see

Sect. 5.4.1). In most of these experiments an adsorbate is dragged with the tip due

to an attractive force between the foremost tip apex atoms and the adsorbate. By

adjusting a large or a small tip–surface distance via the tunneling resistance, it is

possible to switch between imaging and manipulation. Recently, it has been demon-

strated that controlled manipulation of individual atoms is also possible in the dy-

namic mode of atomic force microscopy, i.e., FM-AFM. Vertical manipulation was

demonstrated by pressing the tip in a controlled manner into the Si(111)-(7×7) sur-

face [149]. The strong repulsion leads to the removal of the selected silicon atom.

The process could be traced by recording the frequency shift and the damping sig-

nal during the approach. For lateral manipulation a rubbing technique has been uti-

lized [150], where the slow scan axis is halted above a selected atom, while the

tip–surface distance is gradually reduced until the selected atom hops to a new sta-

ble position. Figure 5.24 shows a Ge adatom on Ge(111)-c(2×8) that was moved

during scanning in two steps from its original position (a) to its ﬁnal position (c). In

fact, manipulation by FM-AFM is reproducible and fast enough to write nanostruc-

tures in a bottom-up process with single atoms [151].

5.5.4 Electrostatic Force Microscopy

Electrostatic forces are readily detectable by a force microscope, because the tip

and sample can be regarded as two electrodes of a capacitor. If they are electrically

connected via their back sides and have diﬀerent work functions, electrons will ﬂow

5 Low-Temperature Scanning Probe Microscopy 219

between the tip and sample until their Fermi levels are equalized. As a result, an

electric ﬁeld and, consequently, an attractive electrostatic force exists between them

at zero bias. This contact potential diﬀerence can be balanced by applying an ap-

propriate bias voltage. It has been demonstrated that individual doping atoms in

semiconducting materials can be detected by electrostatic interactions due to the

local variation of the surface potential around them [152,153].

Detection of Edge Channels in the Quantum Hall Regime

At low temperatures, electrostatic force microscopy has been used to measure the

electrostatic potentialin the quantum Hall regimeof a two-dimensional electron gas

(2-DEG) buried in epitaxially grown GaAs/AlGaAs heterostructures [154–157]. In

the 2-DEG, electrons can move freely in the x–y-plane, but they cannot move in

z-direction. Electrical transport properties of a 2-DEG are very diﬀerent compared

to normal metallic conduction. Particularly, the Hall resistance R

= h/ne

(where

h represents Planck’s constant, e is the electron charge, and n = 1, 2,...) is quan-

tized in the quantum Hall regime, i.e., at suﬃciently low temperatures (T < 4K)

and high magnetic ﬁelds (up to 20 T). Under these conditions, theoretical calcu-

lations predict the existence of edge channels in a Hall bar. A Hall bar is a strip

conductor that is contacted in a speciﬁc way to allow longitudinal and transversal

transport measurements in a perpendicularmagnetic ﬁeld. The current is not evenly

distributed over the cross section of the bar, but passes mainly along rather thin

paths close to the edges. This prediction has been veriﬁed by measuring proﬁles

of the electrostatic potential across a Hall bar in diﬀerent perpendicular external

magnetic ﬁelds [154–156].

Figure5.25a shows theexperimentalset-up usedto observe these edge channels

on top of a Hall bar with a force microscope. The tip is positioned above the surface

Tip position (μm)

212

46810

B (T)

SFM

tip

AlGaAs

GaAs 2DEG

a) b)

Fig. 5.25. Conﬁguration of the Hall bar within a low temperature (T < 1 K) force micro-

scope (a) and proﬁles (y-axis) at diﬀerent magnetic ﬁeld (x-axis) of the electrostatic potential

across a 14-µm-wide Hall bar in the quantum Hall regime (b). The external magnetic ﬁeld

is oriented perpendicular to the 2-DEG, which is buried below the surface. Bright and dark

regions reﬂect the characteristic changes of the electrostatic potential across the Hall bar at

diﬀerent magnetic ﬁelds and can be explained by the existence of the theoretically predicted

edge channels (courtesy of E. Ahlswede; cf. [156])

220 Markus Morgenstern et al.

of a Hall bar under which the 2-DEG is buried. The direction of the magnetic ﬁeld

is oriented perpendicular to the 2-DEG. Note that, although the 2-DEG is located

several tens of nanometers below the surface, its inﬂuence on the electrostatic sur-

face potential can be detected. In Fig. 5.25b, the results of scans perpendicular to

the Hall bar are plotted against the magnitude of the external magnetic ﬁeld. The

value of the electrostatic potential is grey-coded in arbitrary units. In certain ﬁeld

ranges, the potential changes linearly across the Hall bar, while in other ﬁeld ranges

the potential drop is conﬁned to the edges of the Hall bar. The predicted edge chan-

nels can explain this behavior. The periodicity of the phenomenon is related to the

ﬁlling factor ν, i.e., the number of Landau levels that are ﬁlled with electrons (see

also Sect. 5.4.4). Its value depends on 1/B and is proportional to the electron con-

centration n

in the 2-DEG (ν = n

h/eB,whereh represents Planck’s constant and

e the electron charge).

5.5.5 Magnetic Force Microscopy

To detect magnetostatic tip–sample interactions with magnetic force microscopy

(MFM), a ferromagnetic probe has to be used. Such probes are readily prepared by

evaporating a thin magnetic layer, e.g., 10nm iron, onto the tip. Due to the in-plane

shape anisotropy of thin ﬁlms, the magnetization of such tips lies predominantly

along the tip axis, i.e., perpendicular to the surface.Since magnetostatic interactions

are long-range, they can be separated from the topography by scanning in a certain

constant height (typically around 20nm) above the surface, where the z-component

of the sample stray ﬁeld is probed (see Fig. 5.26a). Therefore, MFM is always op-

erated in noncontact mode. The signal from the cantilever is directly recorded while

the z-feedback is switched oﬀ. MFM can be operatedin the static mode or in the dy-

namic modes (AM-MFM at ambient pressures and FM-MFM in vacuum). A lateral

resolution below 50nm can be routinely obtained.

Observation of Domain Patterns

MFM is widely used to visualize domain patterns of ferromagneticmaterials. At low

temperatures, Moloni et al. [158] observed the domain structure of magnetite below

its Verwey transition temperature (T

= 122K), but most of the work concentrated

on thin ﬁlms of La

1−x

MnO

[159–161]. Below T

, the conductivity decreases

bytwo ordersof magnitudeand a smallstructuraldistortionis observed.The domain

structure of this mixed-valence manganite is of great interest, because its resistivity

strongly depends on the external magnetic ﬁeld, i.e., it exhibits a large colossal-

magnetoresistive eﬀect. To investigate the ﬁeld dependence of the domain patterns

under ambient conditions, electromagnets have to be used. They can cause severe

thermal drift problems due to Joule heating of the coils by large currents. Fields

on the order of 100mT can be achieved. In contrast, much larger ﬁelds (more than

10T) can be rather easily produced by implementing a superconducting magnet in

low-temperature set-ups. With such a design, Liebmann et al. [161] recorded the

5 Low-Temperature Scanning Probe Microscopy 221

400 mT

800 mT

0 mT

205 mT

360 mT

a) b) c)

Fig. 5.26. Principle of MFM operation (a) and ﬁeld-dependent domain structure of a ferro-

magnetic thin ﬁlm (b)–(f) recorded at 5.2 K with FM-MFM. All images were recorded on

the same 4 µm×4 µm scan area. The La

0.7

0.3

MnO

/LaAlO

system exhibits a substrate-

induced out-of-plane anisotropy. Bright and dark areas are visible and correspond to attrac-

tive and repulsive magnetostatic interactions, respectively. The series shows how the domain

pattern evolves along the major hysteresis loop from, i.e., zero ﬁeld to saturation at 600 mT

and back to zero ﬁeld

domain structure along the major hysteresis loop of La

0.7

0.3

MnO

epitaxially

grownonLaAlO

(see Fig. 5.26b–f). The ﬁlm geometry (the thickness is 100nm)

favors an in-plane magnetization,but the lattice mismatch with the substrate induces

an out-of-planeanisotropy. Thereby, an irregular pattern of strip domains appears at

zero ﬁeld. If the external magnetic ﬁeld is increased, the domains with antiparallel

orientation shrink and ﬁnally disappear in saturation (see Fig. 5.26b,c). The residual

contrast in saturation (d) reﬂects topographic features. If the ﬁeld is decreased after

saturation(see Fig.5.26e,f),cylindricaldomainsﬁrst nucleateand then start to grow.

At zero ﬁeld, the maze-type domain pattern has evolved again. Such data sets can

be used to analyze domain nucleation and the domain growth mode. Moreover, due

to the negligible drift, domain structure and surface morphology can be directly

compared, because every MFM can be used as a regular topography-imaging force

microscope.

Detection of Individual Vortices in Superconductors

Numerous low-temperature MFM experiments have been performed on supercon-

ductors [162–169]. Some basic features of superconductors have been mentioned

already in Sect. 5.4.5. The main diﬀerence of STM/STS compared to MFM is its

high sensitivity to the electronicproperties of the surface. Therefore, careful sample

preparation is a prerequisite. This is not so important for MFM experiments, since

the tip is scanned at a certain distance above the surface.