Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2

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1090 U. Weierstall

be shown that the underlying atomic lattice gives rise to additional

features in reciprocal space arising from coherent interference with

surface state-derived Bloch waves.

3.1.3 Electron Conﬁ nement

When electrons are conﬁ ned to length scales close to the de Broglie

wavelength, their behavior is dominated by quantum mechanical

200

100

–100

0.00 0.05 0.10 0.15 0.20

k [Å

–1

]

E [meV]

dI/dV [arb.units]

–100 0 100

–65 meV

= (–65  3) meV

m* = (0.40

 0.01) m

Figure 17–15. Energy dispersion relation of the Ag(111) surface state obtained

from the measurement shown in Figure 17–14. The solid line is a quadratic ﬁ t

to the STM data and the dotted line shows results from photoemission data

at 65 K (Paniago et al., 1995). The inset shows a dI/dV spectrum taken on a

perfect terrace (V = 497 mV, I = 5.0 nA) showing the onset of the surface state

at 65 mV below the Fermi energy. (From Jeandupeux et al., 1999.)

Figure 17–16. Direct visualization of the surface state dispersion E(k). (a)

Experimental dI/dV data from Ag(110) at 4 K as a function of energy and dis-

tance from a monoatomic step. (b) One-dimensional Fourier transform of the

dI/dV vs. Y data, which is a direct visualization of the surface state disperison.

(c and d) Results from calculations. (From Pascual et al., 2001b.)

Chapter 17 Low-Temperature Scanning Tunneling Microscopy 1091

effects, i.e., discrete energy levels are formed. The locality of the probe

in STM has an advantage over ARPES for the measurement of the

effects of lateral conﬁ nement of surface-state electrons on the surface-

state energies. Theoretical calculations have suggested that lateral con-

ﬁ nement raises the energies of the surface-state electrons, resulting in

a depopulation of the surface-state band and a modiﬁ cation of surface

properties associated with the surface-state electrons (Bertel et al., 1995;

Memmel and Bertel, 1995; Garcia and Serena, 1995). The energy levels

of surface-state electrons conﬁ ned within artiﬁ cial nanostructures

have been measured with an LT-STM for the ﬁ rst time by Crommie

et al. (1993a). By manipulation with the STM tip, they positioned 48 Fe

atoms on Cu(111) in the form of a circle, forming a so-called quantum

corral, which conﬁ nes surface-state electrons in the inner area (Figure

17–17). A standing wave pattern was observed within the corral. dI/dV

spectra taken in the center showed that the energy levels of the con-

ﬁ ned electrons are discrete. Good agreement with quantum mechani-

cal calculations of the energy levels for a particle in a box was found.

For ideal conﬁ nement of electrons (i.e., inﬁ nitely high potential walls)

Figure 17–17. Various stages during the construction of a circular quantum

corral consisting of 48 Fe atoms on Cu(111) (measured at 4 K). (From Crommie

et al., 1993a. Reproduced by permission of IBM Research, Almaden Research

Center.)

1092 U. Weierstall

the energy levels E

should be inﬁ nitely sharp apart from instrumental

contributions. The measured peak width was interpreted as a lifetime

effect via ∆t

≈

h/∆E

, dominated by scattering into bulk states (see

Section 3.1.4).

Another experiment with a 76-atom “stadium” by the same group

(Heller et al., 1994) gave further information about the scattering mech-

anism of surface-state electrons at the Fe adatoms. Comparison with s-

wave multiple scattering theory showed that the Fe atoms reﬂ ect only

about 25% of the incident electron wave, while 25% are transmitted and

50% are absorbed. Absorption is related to scattering into bulk states.

More recently a systematic study of electron conﬁ nement on nanoscale

hexagonal Ag islands on Ag(111) was performed (Li et al., 1998b). In

contrast to adatom corrals, these are stable structures even at elevated

bias voltages in the STM. Figure 17–18 shows a series of dI/dV maps

taken on a hexagonal Ag island on Ag(111). For voltages around −65 mV

and lower (tunneling form of the sample) the image of the interior of

the island is featureless, since this is below the onset of the surface state

at −67 mV (Li et al., 1997). At higher voltages standing wave patterns are

observed, which originate from oscillations in the LDOS of surface-

state electrons, conﬁ ned by the rapidly rising potential at the edges of

the island. By recording local dI/dV spectra above the center of an island

with an open feedback loop and varying tunnel voltage, peaks at dif-

ferent energy where recorded, which correspond to the energy levels of

the conﬁ ned surface-state electrons. The validity of the particle in a box

model for these conﬁ ned surface states was conﬁ rmed. Analyzing the

measured energy levels in terms of a particle in a box model showed

that the energies conform to the expected scaling behavior down to the

smallest island sizes. The experiments conﬁ rmed that lateral conﬁ ne-

ment is a mechanism for surface-state depopulation since it causes a

discretization of the formerly continuous SS band and an upward shift

of the surface-state levels. The fact that ARPES on vicinal Cu(111) sur-

faces (Sanchez et al., 1995) failed to observe the shifts in the surface state

Figure 17–18. Electron conﬁ nement on nanoscale hexagonal Ag islands. Upper row: constant-current

image of a hexagonal Ag island on Ag(111) (area ∼94 nm

) and a series of dI/dV maps recorded at 50 K

at different bias voltages as indicated. Lower row: calculated local density of states for the hexagonal

box shown at the left. (From Li et al., 1998b.)

Chapter 17 Low-Temperature Scanning Tunneling Microscopy 1093

energies expected on the basis of ideal conﬁ nement is most likely due

to the presence of a sizable distribution of terrace widths within the spot

area of the incident radiation.

Electron conﬁ nement effects have also been observed in metallic

single-walled carbon nanotubes (SWCNTs) (Maltezopoulos et al., 2003).

With an LT-STM operating at 14 K (Pietzsch et al., 2000b), dI/dV spectra

and maps on SWCNTs have been recorded (Figure 17–19). Peaks close

to the Fermi energy are restricted to a certain area of the nanotube and

have been assigned to conﬁ ned states within the SWCNT created by

defect scattering (see Figure 17–20). This shows that defects can lead

to signiﬁ cant backscattering within individual metallic SWCNTs. These

ﬁ ndings are important since SWCNTs are considered promising can-

didates for molecular electronics.

LT-STM measurements of the two-dimensional structure of indi-

vidual electron wavefunctions in SWCNTs have been reported (Lemay

et al., 2001). An SWCNT was deposited on an Au(111) surface. To

increase the electronic energy-level spacing, the nanotube was cut to

less than 40 nm length by applying short bias voltage pulses between

tip and sample. Local spectroscopy with the tip above the nanotube

showed an energy spectrum with a series of sharp peaks, representing

conﬁ nement-induced energy levels. Differential conductance maps

measured at the energies of the peaks revealed spatial patterns that

could be understood from the electronic structure of a single graphite

sheet. The patterns observed are a representation of individual molecu-

lar wavefunctions |ψ

(r)|

. The wavefunctions exhibited “beating”

between electron waves with slightly different wavevectors, and this

intereference effect was exploited to directly probe the electronic dis-

persion relation of an individual nanotube.

3.1.4 Lifetime Measurements

Improved energy resolution (∼3 meV) in photoemission spectroscopy

has made it possible to study the lifetimes of photoholes in a surface

Figure 17–19. Electron conﬁ nement in single wall carbon nanotubes: dI/dV

spectrum of a metallic SWCNT. The oscillations around the Fermi energy (0 V)

are shown in more detail in Figure 17–20. Inset shows atomically resolved STM

image of SWCNT at 14 K. (From Maltezopoulos et al., 2003.)

1094 U. Weierstall

Figure 17–20. (a) STM

image of an SWCNT

end. (b) STS data

(∼LDOS) taken on the

left-hand side of the

dotted line in (a). (c) STS

data taken on the right-

hand side of the dotted

line in (a). (d) Spatially

resolved dI/dV map

taken at the energy of

the 46 mV peak

shows conﬁ nement of

these states to the area

on the left. (From

Maltezopoulos et al.,

2003.)

state. Furthermore, femtosecond time-resolved two photon photoemis-

sion (2PPE) experiments opened up a new path to measure lifetimes

of hot electrons in metals. A hole in an energy band or a hot electron

are quasiparticles, elementary excitations of an interacting Fermi liquid.

A quasiparticle has a lifetime that is the duration of the excitation. In

combination with its velocity, the lifetime determines the mean free

path of the quasiparticle and thereby the range of inﬂ uence of the

Chapter 17 Low-Temperature Scanning Tunneling Microscopy 1095

excitation. Lifetimes are determined by inelastic scattering processes,

i.e., electron–electron (e–e) scattering and electron–phonon (e–ph) scat-

tering. Fermi liquid theory predicts an inverse second power law for

the energy dependence of the e–e lifetime. It also predicts that e–e

scattering dominates e–ph scattering at low temperatures and large

excess energies (≥0.5 eV), whereas at energies very close to the Fermi

level inelastic scattering is dominated by e–ph processes at all tempera-

tures (Pines, 1966).

In metallic systems, quasiparticle states in two-dimensional systems

are important in surface science (Memmel, 1998) and nanoscale tech-

nology (Himpsel et al., 1998). One example for a two-dimensional

systems is a surface state on a noble metal, and here the L-gap Shock-

ley-type surface state on the (111) surface of noble metals has played a

special role as model systems for the investigation of two-dimensional

electron systems. The lifetime of a hole in such a surface state band has

been extensively studied both theoretically and experimentally

(Matzdorf, 1998), but the discrepancy between experiment and theory

remained large. This lifetime is of interest as it determines the effective

range of surface-mediated interactions and is important for epitaxial

growth and chemical reactions at surfaces (Memmel and Bertel, 1995;

Bertel et al., 1995). Theoretical models predicted signiﬁ cantly longer

lifetimes than were observed with photoemission spectroscopy. Using

Figure 17–21. Imaging individual molecular wavefunctions in a carbon nano-

tube at 4.6 K. (a) Constant-current image of metallic SWNT cut to a length of

34 nm by bias voltage pulses (scale bar 10 nm). (b) High-resolution constant-

current image of the shortened nanotube (scale bar 0.5 nm). (c) Differential

conductance spectrum (LDOS) averaged over the area shown in (b) showing

particle in a box states. (d–f) Differential conductance maps showing individ-

ual states at −95, 30, and 96 meV. (g–i) Calculated spatial maps of individual

molecular wavefunctions. The characteristic features of each image are well

reproduced. (From Lemay et al., 2001.)

1096 U. Weierstall

ARPES from a surface state, the lifetime of a photohole state can be

determined by analyzing the spectral linewidth Γ of a photoemission

excitation. The spectral linewidth of a photoemission line gives access

to the lifetime τ via Γ =

h/τ. ARPES integrates over a large surface area

(typically ≥1 mm

) and contains contributions from surface imperfec-

tions over the whole investigated surface range (Theilmann et al., 1997).

Therefore peak widths in ARPES are inﬂ uenced by surface conditions

(defects, roughness) as well as ﬁ nite energy and angular resolution

(so-called nonlifetime effects). Defects on the surface are known to

strongly couple the surface-state electrons to bulk states and thus

reduce lifetimes (Crampin et al., 1994). It would be desirable to apply

the concept of measuring lifetimes from the linewidth of spectral fea-

tures directly to STS. The nonlifetime effects could then be avoided by

measuring the lifetime locally with an STM tip on a spot bare of impu-

rities. The ability of the STM to detect surface topography and to iden-

tify minute amounts of contamination well below the limits of

conventional surface analytical techniques ensures that an effectively

defect-free surface is studied. Since STS does not offer momentum

resolution, measuring the lifetime of a speciﬁ c state deﬁ ned by its

momentum k

and energy E may seem impossible. This problem can

be circumvented to some extend as shown below.

The surface-state (excited hole) lifetime on Ag(111) has been deter-

mined at the dispersion minimum (k

= 0) by evaluating the linewidth

of the surface state onset in dI/dV tunneling spectra (Li et al., 1998a;

Pivetta et al., 2003; Kliewer et al., 2000). Calculated dI/dV spectra show

(Li et al., 1998a) that this linewidth depends on the imaginary part Σ

of the electron self-energy. As Σ increases, the onset of the surface-state

contribution is seen to broaden. Therefore the measured linewidth can

be used to estimate Σ or the lifetime τ =

h/2Σ (Li et al., 1998a) of a hole

in the surface state at the band minimum. The ﬁ rst lifetime measure-

ments (Li et al., 1998a) were performed at a temperature of 5 K using

W tips on Ag(111). Differential conductance spectra were recorded

under open feedback loop conditions, adding a modulation voltage

(1–10 mV at ω = 230 Hz) and using a lock-in ampliﬁ er to record the

signal at ω. Figure 17–22 shows a dI/dV spectrum measured on a large

defect-free terrace showing the characteristic surface state band onset

of width ∆, used to estimate the spectral linewidth (also called lifetime

width or inverse lifetime) Γ =

h/τ. The geometric width ∆ is a combina-

tion of the spectral linewidth Γ of the surface state, thermal broaden-

ing, and additional broadening due to the modulation technique used

to measure the spectrum. The electron self-energy Σ and the corre-

sponding lifetime τ = 67 ± 8 fs were determined from a line-shape

analysis. This measured spectral linewidth was considerably smaller

than previous PES measurements. A more recent lifetime measure-

ment (Pivetta et al., 2003) with the same method resulted in even

smaller spectral linewidths, which were about 1 meV smaller than

recent photoemission (Nicolay et al., 2000) results and theoretical pre-

dictions (Eiguren et al., 2002). Since STS is measured on locally defect-

free surface regions, a smaller spectral linewidth (i.e., a larger lifetime

of surface hole states) than in photoemission is expected. Lifetimes

Chapter 17 Low-Temperature Scanning Tunneling Microscopy 1097

resulting form STS lineshape analysis where τ = 120, 35, and 27 fs for

excited holes at the surface-state band edge on Ag, Au, and Cu, respec-

tively (Kliewer et al., 2000). A comparison of linewidths obtained from

STM measurements with previous PES data (Kliewer et al., 2000)

showed that for Ag(111) and Au(111), STM measures linewidths smaller

by a factor of 3 compared to PES (i.e., longer lifetimes). This illustrates

how large the contribution from electron scattering at surface imper-

fections is. Cu(111) extrapolation of the PES linewidth to zero defect

density (Theilmann et al., 1997) resulted in good agreement with the

STM value, which provides evidence that STS measurements avoid

defect-induced broadening of the linewidth.

In the absence of defect scattering, two processes limit the lifetime

of the hole: inelastic e–e scattering and e–ph coupling. Inelastic e–e

scattering results in the hole being ﬁ lled by interband transitions

involving bulk electrons near the surface or by intraband transitions

involving surface-state electrons (Figure 17–23). These contributions

have been estimated theoretically and good agreement of STS mea-

surements of the lifetime width with theoretical values has been

achieved (Kliewer et al., 2000).

With the method described, lifetimes can be studied only at a single

energy, namely the energy E

at the surface band minimum. Therefore,

in a different approach, the electron lifetimes have been measured with

an LT-STM (Burgi et al., 1999; Jeandupeux et al., 1999; Burgi et al., 2000a;

Vitali et al., 2003) by studying the decay of quantum mechanical inter-

ference patterns from surface-state electrons scattering off step edges.

The decay length is inﬂ uenced by the loss of coherence and hence the

phase relaxation length. The phase-relaxation length, i.e., the distance

an electron can travel without losing its phase information, is directly

related to the lifetime. If the lifetime is shorter than the time necessary

Figure 17–22. Surface state lifetime

measurement. dI/dV spectrum mea-

sured on a large defect-free terrace on

Ag(111), showing steplike onset of the

surface state near −70 mV. The surface

state lifetime can be determined from

the width ∆ by ﬁ tting it to calculated

spectra. (From Li et al., 1998a.)

1098 U. Weierstall

to reﬂ ect the electron wave back from the step, the phase information

is lost and the electron wave cannot interfere with itself anymore. The

electron lifetime τ is related to the phase relaxation length by τ = L

/ν

where ν

is the group velocity (Datta, 1995). The LDOS oscillation

induced by scattering at a surface step is intrinsically damped and its

intensity falls off like

1 kx

, where k is the electron wave vector and

x the distance from the step. The phase relaxation length leads to an

additional damping e

−x/L

of the interference pattern and can be deter-

mined by ﬁ tting a model function to the data. L

(E) and thus τ(E) could

be extracted from dI/dV scans across step edges for the Shockley-type

surface states on Ag(111) and Cu(111). This method has the advantage

that quasiparticle lifetimes can be measured as a function of excess

energy and not only at the surface-state band edge Γ

as with the STS

peak width method described by Li et al. (1998a). Lifetimes of hot

electrons injected by the STM tip have been measured for energies

from the Fermi level to 3 eV excess energy (Vitali et al., 2003; Burgi

et al., 1999). Figure 17–24 shows a schematic energy diagram of the

tunneling process: an electron tunnels from the Fermi level of the tip

into an unoccupied level of the metal, from which, after a short time,

it will decay to the ground state. Experimental lifetimes of hot elec-

trons as a function of excess energy are shown in Figure 17–25. These

energy-dependent lifetime measurements of hot electrons have been

interpreted in terms of e–e scattering, since e–ph lifetimes are inde-

pendent of energy and exceed the measured lifetimes considerably.

Figure 17–23. Schematic energy diagram of electron (solid circles) tunneling

from the bottom of the occupied surface state band of the sample to a metallic

tip (sample bias negative with respect to the tip). Sample states are shown as

a projected band structure. The dashed parabola indicates the maximum of

the bulk bands. The intrinsic lifetime of the hole left behind by an electron

tunneling from sample to tip is determined by inelastic electron–electron

scattering, which results in either an intraband transition (two-dimensional)

or an interband transition (three-dimensional). (From Kliewer et al., 2000.)

Chapter 17 Low-Temperature Scanning Tunneling Microscopy 1099

The experimental data are found to scale with (E − E

)

−2

above 0.5 eV

as predicted by Fermi-liquid theory, but deviate from the quadratic

scaling law at low energies. Calculations showed that the main scat-

tering process at low energies are inelastic intraband e–e scattering and

e–ph scattering, whereas at higher energies e–e interband scattering

with bulk electrons dominates.

To determine the e–ph scattering contribution to the lifetime,

temperature-dependent lifetime measurements have been done with

low-energy quasiparticles (Jeandupeux et al., 1999). Constant-current

2.0

1.5

1.0

0.5

0.0

–0.5

0.1 0.2 0.3

Tip

Ag(111)

Intra

Iater

(Å

–1

)

(E-E

) (eV)

Figure 17–24. Schematic energy diagram of the tunneling process of hot elec-

trons into the surface state band (sample bias positive with respect to the tip).

Dots indicate experimental measurements in Vitali et al. (2003). Electrons

injected at an energy E scatter inelastically with other electrons into unoccu-

pied states with an energy smaller than E. (From Vitali et al., 2003.)

Figure 17–25. Experimental lifetime of hot surface state electrons at the

Ag(111) surface as a function of energy. The solid line is a ﬁ t of the high energy

data (solid squares) to the expected (E − E

)

−2

behavior in the Fermi liquid

model. (From Vitali et al., 2003.)