nanowires on germanium using angle
resolved photoemission spectroscopy
Author:
Pieter Rens Pronk
Supervisors: N. de Jong, M.Sc. prof. dr. M. S. Golden Second Supervisor: dr. K. Dohnalov´a
Physics M.Sc. research project thesis
Track: Master Advanced Matter and Energy Physics van der Waals-Zeeman Institute
One dimensional metallic systems are predicted to show exotic physics like spin-charge separation. One way to experimentally access such systems is by self-assembling growth of metallic atoms into nanowires on semiconductor sub-strates. In this project, highly ordered germanium surfaces were prepared and used as a template for the generation of well-separated gold nanowires with lengths exceeding 100nm. Germanium surfaces were cleaned by argon bombard-ment and flash annealing in ultra-high vacuum before gold was evaporated on these surfaces. Surface reconstructions were observed before and after gold de-position via low energy electron diffraction (LEED). Angle resolved photoemis-sion spectroscopy (ARPES) was chosen as a direct way to measure the electronic band structure of these surface structures. We are the first to perform ARPES investigations of gold nanowires grown on the Ge(110) surface, which is unique for its single domain wire growth mode. Lab-based ARPES data from highly or-dered Ge(110) revealed gold-induced umklapp-related features in the electronic band structure. These features match the LEED findings that indicated a short-ening of the surface Brillouin zone in one direction. A set of nanowire samples on both Ge(110) and Ge(100) surfaces was transported in an ultrahigh-vacuum container to the BESSY-II synchrotron in Berlin. These samples were used for photon energy dependent ARPES investigations (20eV< hν < 120eV). This was the first successful transfer of nanowire samples from the Amsterdam lab to the synchrotron. The ARPES data showed that the observed features in the elec-tronic structure were dispersive in kz, revealing their bulk electronic character.
A subsequent scanning tunneling microscopy (STM) study carried out at the University of Twente on the same samples [1] indicated that a heat treatment process was necessary - after the transport from one lab to the next - to remove an absorbate layer that masked the nanowire structures. Therefore the lack of clearly nanowire related ARPES signals reported here, plus the STM data after annealing, point to the necessity of further surface treatment to remove adsorbate overlayers prior to future ARPES experiments.
1. Introduction . . . 5
2. Theory . . . 7
2.1 Fermi liquid . . . 7
2.2 Breakdown of Fermi liquid theory . . . 9
2.3 Tomonaga Luttinger Liquid (TLL) . . . 12
2.4 TLL physical observables . . . 13 3. Methods . . . 15 3.1 Sample preparation . . . 15 3.1.1 Sputtering . . . 16 3.1.2 Annealing . . . 17 3.1.3 Gold evaporation . . . 17 3.2 Surface analysis . . . 18
3.2.1 Photoemission spectroscopy (PES) . . . 18
3.2.2 The work function . . . 19
3.2.3 Angle resolved photoelectron spectroscopy (ARPES) . . . 19
3.2.4 X-ray Photoemission Spectroscopy (XPS) . . . 23
3.2.5 Inelastic mean free path (IMFP) . . . 24
3.2.6 Surface states . . . 25
3.2.7 FAMoS . . . 27
3.2.8 BESSY-II storage ring, One-Squared end station . . . 29
3.2.9 UHV Suitcase . . . 30
3.2.10 Low energy electron diffraction (LEED) . . . 31
3.2.11 LEED pattern analysis . . . 32
3.2.12 Scanning tunneling microscopy (STM) and spectroscopy (STS) . . . 34 4. Materials . . . 36 4.1 Bulk germanium . . . 36 4.2 Ge(100) . . . 37 4.3 Ge(110) . . . 38 4.4 Au superstructures on Ge(100) . . . 40 4.5 Au superstructures on Ge(110) . . . 42
5. Results and discussion . . . 44
5.1 Results on Ge(100) and Ge(100)/Au . . . 44
5.1.1 LEED results on Ge(100) and Ge(100)/Au . . . 44
5.1.2 ARPES results on Ge(100)/Au . . . 46
5.2.1 LEED results on Ge(110) and Ge(110)/Au . . . 49
5.2.2 ARPES results on Ge(110) and Ge(110)/Au . . . 51
5.3 XPS results on Ge(110), Ge(110)/Au and Ge(100)/Au . . . 57
6. Summary . . . 61
7. Outlook . . . 63
8. Acknowledgements . . . 68
One dimensional (1D) systems are often used as simple model systems to ex-plain more complex materials at higher dimension. However 1D physics by itself is of great importance as well since electronic techniques are rapidly moving to-wards the single-atomic scale. As an indication of current developments: in 2020 a typical semiconductor integrated circuit node is expected to measure 5nm, that corresponds to roughly 10 atomic layers [2]. Moreover, 1D metallic systems are predicted to display completely different behavior then their higher dimensional counterparts. A theoretical framework for 1D metallic systems was developed in the 1960s by Tomonaga and Luttinger but experimental research still lags behind. Therefore experimental investigations of 1D systems are in-teresting not only because of the electronic applications but also because of their experimentally unresolved physical properties. In this project angle re-solved photoemission spectroscopy (ARPES) was used to study the properties of possibly 1D metallic systems. ARPES is an extremely surface sensitive tech-nique and it is therefore ideally suited to study 1D metallic wires arranged on a surface. In the last two decades the race for the experimental investigation of 1D systems was focused on ARPES investigations of noble metal nanowires on semiconducing surfaces. There is an ongoing debate about the ARPES studies on Ge(100)/Au systems that have been performed. With the work behind this thesis, we join the field of the experimental investigation of 1D metallic systems on germanium surfaces. Data is reported from Ge(100)/Au and a novel germa-nium based system with Miller indices (110) that appears to have exceptional qualities compared to Ge(100)/Au is introduced.
Recently, two groups successfully grew 1D gold nanowires on Ge(100) and in-vestigated their electronic structures using ARPES [3, 4]. Their reports partially agree on the findings as regards the surface topography, but disagree strongly as regards the dimensionality of the electronic states. Systems like Ge(100)/Au are known to yield nanowire structures existing in two orthogonal domains. This property severely complicates the analysis of ARPES data from Au on Ge(110), since information on the electronic structure is then a superposition of two wire orientations. After this master’s research project was started, the group of Zandvliet in Twente discovered a novel 1D system that was never stud-ied before: Ge(110)/Au. STM images of the surface revealed 1D wires pointing all in the same direction, making this potentially an ideal candidate to examine for 1D behavior in ARPES. We created and studied both the Ge(100)/Au and Ge(110)/Au 1D systems for this project.
There are of course other ways to create 1D electronic systems and to ob-serve these. Carbon nanotubes for example are expected to host 1D electronic behavior. Carbon nanotubes are essentially honeycomb structured 2D sheets of carbon atoms, known as graphene, rolled into a tube. The honeycomb, 2D graphene sheets have become a rich field of research into 2D electron systems.
However, when such a sheet is rolled into a tube, the k values in circumferential direction are quantized [5]. The result is that only certain k values are allowed, which cuts the original 2D bands structure into 1D bands. A second example of 1D systems are cold rubidium gases confined to 1D by a magnetic field. In such a setup the motion of the rubidium atoms, that are bosons instead not fermions, is of interest [6]. Their character is indirectly observed via the interference pat-tern of two colliding 1D gases. One of the major drawbacks is that spin-charge separation is not present in this system because the rubidium atoms are neutral atoms whereas the gold nanowires host charged electrons and therefore might reveal spin-charge separation.
This master’s research project was supervised by N. de Jong and M. S. Golden. The master’s project contributes to de Jong’s PhD project on the spec-troscopic analysis of nanowires. The PhD project is part of a multi-university program titled ”The Singular Physics of 1D electrons” and is funded by the Dutch Foundation for Fundamental Research on Matter (FOM). Collaborating groups at Twente and Amsterdam focus within this program on the experi-mental analysis of the atomic structure and theoretical aspects of such systems respectively. As part of this FOM project, we collaborated with prof. dr. ir. H.J.W. Zandvliet and coworkers (University of Twente), because of their exper-tise on the creation and STM investigation of nanowires.
The master’s project took place in the Quantum Electron Matter (QEM) group, within the Institute of Physics (IoP) of the University of Amsterdam. The QEM group has experience with the creation and investigation of topo-logical insulators (TIs) and superconductors (SCs). This master’s research project kicked off the group’s research on 1D systems, specifically, that on gold nanowires on a germanium template. It required a different approach in terms of creating and transporting the samples with respect to TIs. Therefore the modification of the experimental setup was a significant part of the project.
This thesis starts by pointing out the theoretical differences between one-and higher dimensional metallic systems, thereby introducing the conceptual importance of 1D systems. Secondly, sample preparation will be explained. Then, two types of photoemission spectroscopy techniques that were used to investigate the electronic behavior, and the electron diffraction technique that was used to investigate the surface crystalline structure will be discussed. A brief explanation of scanning tunneling microscopy is given to connect this research to that of Zandvliet et al. The explanation of the methods is followed by an introduction to germanium since this forms the template for the 1D systems. In the final chapter, the results on the Ge(100)/Au and Ge(110)/Au samples are presented and discussed, followed by our conclusion with an outlook towards future research.
2.1
Fermi liquid
Metallic systems in two or more dimensions can be described reasonably well by the Fermi liquid theory. This powerful method will be briefly explained here to introduce the problem that it faces in 1D. The Fermi liquid theory was constructed by Landau in 1956 to describe interacting metallic systems. It de-scribes interactions between nearly free interacting electrons by free fermionic quasiparticles. The quasiparticles represent electrons dressed by density fluc-tuations. Quasiparticles are fermions since they are composed of one electron (fermion) and density fluctuations (bosons). The non-interacting nature of the quasiparticles makes the system easier to describe than the more complicated interacting electron picture. These non-interacting quasiparticles have the same momentum, spin and charge as the original electrons but differ by their mass. Fermi liquid theory successfully describes a vast amount of metals however, not 1D metallic systems. The quasiparticle picture is valid only near the Fermi level due to the following arguments. In general a wavefuction oscillates at a rate determined by the energy of the particle E(k) it describes. This is also valid for quasiparticles. However, quasiparticles have an extra damping term determined by their lifetime τ .
Ψ ∝ e−iE(k)t/¯he−t/τ
The quasiparticle lifetime near the Fermi level becomes relatively large because decay via scattering is prohibited by phase space restrictions. Landau has for-mally shown that the lifetime of quasiparticles in three dimensional systems decreases as the square of the energy away from the Fermi level. On the other hand, the frequency of the quasiparticle wavefunction is relatively large [7] since it is proportional to 1/E(k). For a quasiparticle to be stable, the scattering rate must be smaller than the frequency of the quasiparticle. In other words, the free quasiparticle picture is valid when the scattering rate is small and the quasipar-ticle excitation energy is small, that is near the Fermi level [7].
Although particles described by Fermi liquid theory have charge, spin and momentum in common with a free electron gas their spectral properties diverge at some points. The difference is captured by the spectral function A(k, ω) as a function of frequency ω = E(k)/¯h (Fig. 2.1 (a)(b)). The spectral function A(k, ω) displays the energy distribution of a system when a particle with mo-mentum k is added or removed. Since, in the free case, one single electron excitation does not excite other electrons, a free electron spectral function is given by a delta peak (Fig. 2.1(a)). A quasiparticle spectral function however is, unlike the delta peak of the free electron gas, broadened, meaning other elec-trons are excited as well. The broadening of the quasiparticle spectral function peak scales by the finite lifetime of the quasiparticles. The excitation of other
electrons is a direct consequence of the quasiparticle scattering mechanism il-lustrated above. Since the scattering rate increases away from the Fermi level, the quasiparticle lifetime decreases and the spectral broadening becomes large as shown in figure 2.1(b). A single quasiparticle excitation for an interacting system consists of a single particle excitation that dissipates to other particles thereby determining the excitation lifetime. The occupation factor n(k) near the Fermi level (EF) for a non-interacting fermionic system at T=0 exhibits a
unitary step (Fig. 2.1(c)). This means that at zero temperature all states are occupied up to the Fermi energy and not above the Fermi energy. Moreover the step height, that corresponds to the quasiparticle residue (Z), contains in-formation about the single particle excitation contribution to the quasiparticle. This is expressed via the following quasiparticle perturbative expansion [8].
| ΨN +1 k i = Z 1/2 m c † k| Ψ Ni + 1 V3/2 X k1,k2,k3 δk,k1−k2+k3c † k3ck2c † k1 | Ψ Ni + ...
In this expression, | ΨN +1k i is the quasiparticle state and | ΨNi is the state of
the interacting N-particle system. The bare single particle creation and annihi-lation operators are c†k and ck respectively [8]. The first term on the right side
of the equation therefore corresponds to a single particle excitation. The higher order terms correspond to multiple particle excitations. Near the Fermi level, Zm, which determines the amplitude of the single particle term, approaches the
value Z [8]. If Z is equal to 1, the excitations are described only by single par-ticle excitations, that is the non-interacting case. For a Fermi liquid however, the edge step height of the occupation factor becomes 0 < Z < 1 (Fig. 2.1(d)) meaning that excitations correspond to quasiparticles with some single particle character. However, for 1D systems, Z goes to zero, then the quasiparticle is described by many particle excitations only [9]. In that case, the quasiparti-cles do not correspond to the original single particle anymore [7] and Landau’s approach of the one-to-one correspondence between electrons and quasiparti-cles fails. Moreover, as explained in the next section, only collective bosonic excitations remain in 1D [7, 8].
Fig. 2.1: The spectral function A(ω) corresponding to the Fermi gas (a) and the Fermi liquid (b) Occupation factor n(k) for a Fermi gas (c) and a Fermi liquid (d).
Thus, Fermi liquid theory is insufficient in describing extreme systems such as 1D systems. In 1D, even small interactions become too significant to establish a quasiparticle picture by any means [7]. In the next paragraph it will be further explained that bosonic excitations describe excitations in 1D metallic systems.
2.2
Breakdown of Fermi liquid theory
One would intuitively expect that repulsive interactions between electrons in 1D cause the electrons to push each other forward when propagating, rather then moving around each other (Fig. 2.2). Thus in one dimension an excitation of a single electron becomes a collective excitation. This is in clear contrast to Fermi liquids in higher dimensions. In this paragraph I follow closely the arguments of Giamarchi to explain the breakdown of the Fermi liquid theory more formally [7].
Fig. 2.2: Cartoon of particles confined to 2D (a) and to 1D (b). (Image ref.[7])
I will start by discussing an example of why confined electrons behave differ-ently in 1D. If one would describe electronic interactions perturbatively, a linear response of the electron density can be attributed to the perturbed Hamiltonian. The measure for the response of the system to an external potential (i.e. susceptibility ξ) can be written as [7].
χ(q, ω) = 1 Ω X k fF(ξk) − fF(ξk+q) ω + ξ(k) − ξ(k + q) + iδ
Where fF(ξk) is the Fermi factor as function of energy ξk relative to the
Fermi level. The system volume is written as Ω, ω represents the frequency and δ is a positive small number. A single vector in momentum space Q that causes ξ(k + Q) = ξ(k) = 0 is called a nesting vector. In high dimensional systems nesting is only possible for a very small number of points relative to the overall phase space, whereas for 1D systems such a ”nesting” condition is always met (Fig. 2.3).
Fig. 2.3: Cartoons of Fermi surfaces depicted in 3D k-space. In 1D nesting of the whole surface occurs for just one single vector Q. (Image ref.[7])
When the dispersion is symetric around k = 0, it is clear that nesting occurs if Q = 2kF. For 1D systems nesting always occurs, and therefore the
suscep-tibility, as written above, always diverges. A divergence of a perturbation in general points out that the true ground state has a different character than the starting point of the perturbation theory. If the assumptions of the ground state used by perturbation theory are not reflected in the true ground state then one encounters singularities of the susceptibility. In this case, a different way of describing 1D systems is needed [7].
A second argument for the different behavior of 1D systems is based on the behavior of particle-hole excitations. In these excitations a particle, in our case electron, is excited to above the Fermi level leaving behind a hole. The excited electron gains energy E(q) and momentum q. In a two-dimensional (2D) system it is possible to create a low energy excitation with an arbitrary k between 0 and kF. The excitations are therefore k-dependent. For 1D there are only two
points in k space that cross the Fermi level. Therefore, low energy particle-hole excitations can only exist in the vicinity of these points. This restricts the excitations to be around q = 0 and q = 2kF (Fig. 2.4(a)), whereas in 2D q
can be a continuum between these values (Fig. 2.4(b)). In higher dimensional systems there is a whole branch of available energies and momenta for such an excitation, in 1D there is not (Fig. 2.4).
Fig. 2.4: (a) Particle-hole excitation on a Fermi surface for a 1D system is shown with a particle-hole excitation indicated by a white- and a black dot. The right side displays (shaded) the continuum of particle-hole excitation spectra with the excitation energy versus excitation wave vector. (b) Displays the 2D case.
When the energy of 1D systems is expanded near kF, it can be shown that
the excitation energy for the electron-hole pair only depends on q and that broadening becomes negligible near the Fermi level.
Ek(q) = ξ(k + Q) − ξ(k)
The excitation energy Ek(q) is the difference between the energy of the hole and
the electron. Here a simple quadratic disperion (¯h is set to one) will be used as an illustration, however the reasoning also holds for more complex dispersion relations.
ξ(k) = k
2− k2 F
2m
Focusing only on low lying excitations in the range (kF − q) ≤ k ≤ kF enables
one to write down an average excitation energy E(q) and energy dispersion δE(q) = max(Ek(q)) − min(Ek(q)).
E(q) = kFq m δE(q) = q
2
m
It is clear that the dispersion in energy δE(q) goes to zero faster than the en-ergy E(q) for small excitation momenta q. Thus, in general for 1D systems, δE(q) for particle-hole excitations decays faster than E(q) when the energy of
the excitation approaches the Fermi level. This indicates that the frequency (∝ 1/E(q)) of such a electron-hole excitation becomes larger than the decay rate (∝ 1/δE(q) ). When the decay rate is small the electron-hole excitations become long lived. Thus the particle-hole excitation becomes longer lived near the Fermi level. In analogy to the argument why the quasiparticle description works for Fermi liquids we have now shown that particle-hole excitations are well defined near the Fermi level for 1D systems. Particle-hole excitations are described by creation (particle) and annihilation (hole) operators which, in 1D, together can be rewritten as bosonic density operators. It follows that bosonic charge density fluctuations are dominant near the Fermi level of a 1D metal. When regarding spin systems, the same reasoning results in spin density fluc-tuations. The density and spin fluctuations are uncoupled near the Fermi level and therefore propagate at different velocities [7, 10]. This results in so called spin-charge separation and this can be observed as a band splitting near the Fermi level. An unequivocal experimental observation of spin-charge separation in a 1D electron system would be the highlight of this research on the long term.
2.3
Tomonaga Luttinger Liquid (TLL)
In 1950 Tomonaga realized that electronic excitations in 1D metallic systems could be well described by bosonic excitations [6]. Luttinger introduced a math-ematical framework in 1963 to formally describe bosonization [6]. Haldane gen-eralized the concept in 1981 [7]. The key to describe the system with bosonic density waves is linearization of the spectrum. 1D systems allow linearization near the Fermi level, disregarding the exact details of the entire band structure. The theory describes only low energy excitations of metallic systems, just like the Fermi liquid theory is only valid near the Fermi level. Tomonaga Luttinger liquid theory describes a broad class of 1D confined systems of both fermions and bosons [6]. In this project we focus on fermionic systems.
It is far from trivial to derive an exact expression for the bosonic represen-tation of the Hamiltonian (see Appendix: Bosonization). The spinless Hamilto-nian for a 1D system, expressed using bosonic creation (b†(q)) and annihilation (b(q)) operators, is therefore stated without further ado.
H 'X
q6=0
νF | q | b†(q)b(q)
In this expression, vF is the Fermi velocity and q the momentum of the
ex-citation. The main consequence of bosonization is that the Hamiltonian of interacting fermions is transformed into a Hamiltonian of linear dispersive free bosonic particles. In this way, interactions are much easier to describe because the Hamiltonian in a bosonic basis is diagonal. When electron-electron interac-tions are present, quartic electron creation terms and annihilation terms in the Hamiltonian are introduced. However, since the electrons are described by den-sity operators the interaction can be then written as a product of two bosonic operators instead of a product of four fermionic operators, thereby simplifying the calculation at hand enormously.
2.4
TLL physical observables
As states in last chapter, interacting fermionic particles can be described by free bosonic particles. In this section, we take a closer look at the physical observables of 1D systems and how they differ from higher dimensional systems. Electrons confined to 1D disperse only in one spatial direction. Their group velocity is zero in all other spatial directions. Group velocity is in general proportional to ∂E∂k. Therefore, when looking at the Fermi surface of 1D bands, one typically expects to observe straight lines stretching along the other two spatial directions of k-space as fingerprints of 1D confinement. Such Fermi surface fingerprints do not uniquely describe the system since higher dimensional systems sometimes show similar k-space contours, for example in the shape of a square Fermi surface [7]. Therefore a first, but still inconclusive, indication for 1D nanowire systems is the observation of non-dispersive bands at the Fermi level that are oriented perpendicular to the wires.
A second fingerprint of spinful TLLs is spin-charge separation. As a direct consequence of applying the density operator notation on a spinful 1D system, a decoupled Hamiltonian can be written as:
Hkin= Hσ0+ H 0 ρ.
In this Hamiltonian, the charge and spin degrees of freedom are completely decoupled [7]. Low energy excitations in a TLL manifest themselves for that reason as charge- or spin-density waves called holons and spinons. Both prop-agate with different velocities and therefore induce a splitting of the original spectral function peak near the Fermi level. For a generic interacting 1D elec-tronic system with spin the spectral function contains power-law singularities at the spin-and charge density wave eigenenergies [10]:
A[k, ω = c,s(k)] ∝ [ω − c,s(k)]−µ
c,s(k)
. The exponent µc,s depends on the Luttinger parameter (K
c), which measures
the electron-electron interaction strength. Angle resolved photoelectron spec-troscopy (ARPES), introduced in the next chapter, is a method to experimen-tally observe the spectral function. It would be the triumph of TLL physics to experimentally measure such a band splitting by ARPES.
Another observable that could indicate the existence of a 1D electronic sys-tem is the occupation factor. Fermions occupy all states strictly up to the Fermi level at T = 0, the occupation factor makes a discontinuous step of one at the Fermi level. The occupation of 1D confined electrons, that are fermions, is how-ever completely different from their higher-dimensional counterparts. One of the fingerprint experimental parameters of a TLL is that the occupation factor has no Fermi step at all. Unlike free electrons, a TTL shows no discontinuity at kF but rather a continuous power-law (Fig. 2.5)(a). The occupation factor
n(k), that behaves according to a power-law near kF, can be measured either
by scanning tunneling spectroscopy (STS) or by ARPES.
Though the power-law behavior is unique for 1D systems, it is hard to dis-criminate it from the Fermi step of higher dimensional systems for multiple reasons. The first argument is that a continuous occupation factor for higher dimensional systems can be observed at the Fermi level due to instrumental and temperature broadening [11]. Secondly, an apparent power-law behavior of
the occupation factor for non-TLL systems can be also observed due to a phase transition in nearly 1D systems. One dimensional systems may undergo a tran-sition from metallic to insulating behavior by what is called a Peierls trantran-sition. This is explained by the deformation of a 1D chain of equally spaced atoms into a 1D chain with dimers of atoms (Fig. 2.5(b)&(c)).
Fig. 2.5: (a) Occupation factor for a TLL. (b) Atomic chain in 1D with lattice spacing a. The corresponding band in an extended zone scheme for one valence electron per atomic site is drawn underneath. (c) A Peierls instability is depicted. Atoms move together to form dimers, causing the Brillouin zone to become half. (Image ref. [12])
Such a deformation of the ion lattice increases the lattice energy. However, as a direct consequence of the doubling of the lattice unit cell, the Brillouin zone shortens and a band gap is induced at the Fermi level, which inhibits the electronic states to cross the Fermi level. The bands underneath the Fermi level near the Brillouin zone edges are lowered in energy. Such a Peierls distortion occurs when the total energy of the system, that is the energy of the electrons plus the lattice energy, is lowered.
Peierls distorted systems are, unlike TLL, insulating due to the mechanism described above. However just above the Peierls transition temperature a 1D Peierls system shows a 1D-like power law behavior of the occupation factor [11]. It is therefore not trivial to discriminate a TLL from a Peierls distorted system based on the occupation factor. All these complications mean that highly-challenging spectroscopic experiments are essential in order to catch a glimpse of TLL physics.
3.1
Sample preparation
Sample preparation is one of the key difficulties of experimental studies on 1D systems. In general, studies of the electronic properties on the surface of mate-rials require UHV conditions at all stages between the growth and measurement of the samples. UHV conditions are required because exposure to ambient con-ditions would result in layers of adsorbates on the surface of the sample within a few seconds. If the thickness of the adsorbate layer exceeds the inelastic mean free path (IMFP) of the electrons, inelastic collisions will dominate information about the original momentum and energy of the electrons on a clean surface. Moreover, specifically related to this thesis and Ge-based 1D systems, nanowires can only be created on an adsorbate-free germanium surface such that the crys-talline germanium surface forms a template for the nanowires. In order to take advantage of the self-organization of 1D systems on Ge surfaces, in-situ sample preparation was essential. To this end, the laboratory setup of the QEM group had to be substantially modified. Therefore, this thesis contributed to adding surface science in the expertise of the QEM group, as well as new instrumenta-tion (figure 3.1).
Fig. 3.1: A new preparation chamber, indicated petrol blue, was attached to the already-existing configuration. Due to the specific needs of the samples used in this thesis work a modified manipulator was attached.
cut into 10 × 3mm2 samples. Samples were mounted on a specially-designed
sample holder. The sample holder was cleaned in an ultrasonic ethanol bath before the samples were mounted. The samples were transfered into the UHV chamber via a load-lock. After the sample entered the UHV chamber, a clean surface was obtained by repeating cycles of sputtering and annealing. The crys-talline surface quality was monitored by low energy electron diffraction (LEED) between cycles. This technique reveals the order of a crystalline surface by its reciprocal-space signature. LEED will be explained in more detail later in this chapter. Gold was evaporated when LEED patterns confirmed a highly-ordered crystalline surface. The typical preparation time for optimizing clean samples was 5-8 days, with an average of 6 cycles per day. In the next subsections the preparation techniques will be discussed in more detail.
Fig. 3.2: Image of an Omicron sample holder with the germanium sample clamped in the center. An ear-like handle allows the plates to be transported inside the chamber in UHV conditions. Samples measure 10mm×3mm.
3.1.1 Sputtering
Sputtering is a surface treatment process used to remove the top layers from a sample by bombardment with an accelerated beam of ionized noble gas. In the context of this thesis work, it has been used to clean the sample surfaces from foreign atoms. Inside the sputtering gun, electrons are emitted thermally from a filament and they are accelerated by an anode grid. Collisions of electrons with the noble gas atoms cause ionization of the gas: e−+ A → 2e−+ A+, thereby creating a plasma. At low gas pressure the collision rate is too low to maintain the plasma. On the other hand, very high gas pressure prevents the electrons to pick up energy for the next collision and the ionization does not take place [13]. Therefore, to create a plasma the gas pressure needs to be optimized. Once ionized, the gas ions are accelerated towards the sample. The kinetic energy of the ions is much larger than typical lattice bonding energies; typically in the energy range of 30eV to 1kV. In this range sputtering is dominated by knock-on effects [13]: a chain of surface atom collisions that is induced by the incoming ion. As a result, the atoms on the surface may be detached from the surface. In this thesis work an Omicron SES-10 sputtering gun was used. The emission current was set at 10mA. Ions were accelerated from the source volume towards the sample by E=500eV. A current of 1.8µA through the target sample was induced by ion bombardment. The current was maximized to increase the effective bombardment by focusing the beam with an ion-optic lens. The typical sputtering duration was 40 minutes for the first cleaning cycles and 20 minutes
for subsequent cycles. The sputtering process leaves behind a highly disordered surface structure. After sputtering, the samples are therefore annealed to create a homogeneous crystalline surface as discussed in the next subsection.
3.1.2 Annealing
Annealing is the heating of a sample to remove surface contamination and to rearrange the surface crystalline structure. High thermal energies increase the mobility of the surface atoms and absorbates [14]. The mobility varies strongly with the binding energy of each specific atom. Therefore the effective annealing temperature depends on the specific material and even on the orientation of the materials surface.
The germanium samples were heated by resistive heating in UHV. Specially-designed sample holders by Omicron (Fig. 3.2) enabled resistive heating by forc-ing the current to flow directly through the sample. This is accomplished usforc-ing insulated clamps on the samples. These specific holders required modification of the UHV sample manipulators that allow in-situ transport of the samples from the preparation chamber to the measurement chamber. Resistive heating is a robust way to heat the sample in UHV, minimizing the temperature increase of other UHV elements. Heated objects like the sample plate degas, causing the pressure to rise, thereby inducing contaminations on the sample surface. Heat-ing power of 42W<P<46.8W corresponded to temperatures between 1000K< T<1100K depending on the sample resistance. The resistance of the samples was measured to be typically around R=2 kΩ, corresponding to the resistivity of germanium of 1 Ωm. Samples were heated just below the melting temperature for several seconds depending on the pressure response. Highly contaminated surfaces induced a rapid pressure increase even at relatively low temperatures (T ' 800 K). During annealing, the pressure in the preparation chamber was kept under a pressure of 10−9 mbar. At the final stage of the annealing and sputtering cycles, a sample temperature of 1200K could be maintained up to 10 seconds, with the pressure not exceeding 10−9mbar. Annealing temperatures were measured using a pyrometer. A pyrometer measures the irradiance of a heated object at a specific spectrum. The irradiance (j) is related to temperature (T) and emissivity (ε) as j=εσT4, where σ is the Stefan-Boltzmann constant.
The emissivity (ε) of a material is the ratio of the thermal radiation from that material to the radiation from an ideal black body at the same temperature. The emissivity of germanium at T=900◦C is ε = 0.5, according to literature [15]. Small variations in the value ε and in the angle between the sample and the pyrometer induced a read out temperature error of ±50K. The pyrometer enabled annealing temperatures to be measured in a fast and non-contact way.
3.1.3 Gold evaporation
Once the germanium surface was cleaned, it was used as a template for the self organized growth of the nanowires. In the UHV preparation chamber gold was heated by a tungsten wire up to its melting temperature. Evaporated gold atoms absorb on the germanium surface. After the evaporation process the sample was heated up to 800K to induce self-organization of the gold atoms.
The evaporation rate was calibrated using a quartz microbalance. In brief, the evaporation rate was determined in the following way. The microbalance
was put in front of the evaporator, temporarily replacing the target sample. A piezoelectric quartz microbalance measures the mass per unit area that is deposited on its surface. An alternating current causes lattice vibrations due to the piezoelectric effect. The eigenfrequency depends strongly on the mass. A mass change is thus measured by the shift of eigenfrequency which can be read out as a change in AC current. The microbalance electronic controller used a user-input value for the density of the evaporant which is translated into monolayers. The microbalance thus allows a measure of the evaporation rate that is determined by the gold heating current and the sample exposure time. The measured evaporation rate was 1˚A/167seconds. When the conditions for the desired evaporation rate were determined, the microbalance replaced by the sample.
3.2
Surface analysis
The surface analysis interest is twofold. First, the surface structure needs to be resolved on the atomic scale and the surface compounds need to be identi-fied. Surface analysis techniques that deal with the structural investigation of the surface are scanning tunneling microscopy (STM) and low energy electron diffraction (LEED). The chemical composition of the surface can be identified using X-ray photoemission spectroscopy (XPS). The second point of interest is the electronic band structure of the created systems. The electron dispersion relation is investigated by angle resolved photoemission spectroscopy (ARPES). In the next sections these techniques will be discussed.
3.2.1 Photoemission spectroscopy (PES)
The photoelectric effect forms the basis of PES techniques. The photoelectric effect or Hertz effect, named after its first observer, describes the event of elec-trons that are ejected from a material due to the energy exchange with incoming photons. Hertz was the first to describe that the amount of electric sparks of a spark-gap generator was enhanced when it was illuminated by visible or ultra-violet light [16]. The description of the quantization of light as discrete energy packages - photons - by Planck made it possible for Einstein to ascribe the pho-toelectric effect to the interaction between photons and electrons. Einstein was awarded a Nobel prize in physics for the explanation of the photoelectric effect. In a typical photoemission measurement, monochromatic light, with a spe-cific photon energy (hν), illuminates the sample of interest. An electron in the sample is excited when the photon energy exceeds the binding energy (EB) of
the electron. The excited electron, or photoelectron, is released from the sample if its energy exceeds the work function of the material (φ). Energy conservation dictates that the kinetic energy (Ekin) of the electron outside the sample relates
to the electron binding energy as follows:
Ekin= hν − φ− | EB|
The main objective in PES studies is the mapping of the binding energy versus photoelectron intensity. In this respect, the photon energy is known (property of radiation source) and the photoelectron kinetic energy is measured, only the work function needs to be determined.
Two techniques that were used during this project make use of photoelec-tric effect. First, angle resolved photoemission spectroscopy (ARPES) allows to directly map the electronic dispersion by mapping both kinetic energy and emission angle of the photoelectron. Secondly, photoemission spectroscopy with photon energies in the X-ray regime (XPS) was used to measure the binding energy of the strongly bound core electrons to uniquely identify the chemical composition a material.
3.2.2 The work function
In a PES experiment, the sample and the detector are grounded thereby leveling the Fermi energies of both materials. A potential difference between the work functions of the sample and the detector (∆V = φs− φd) introduces an electric
field which accelerates electrons towards the detector. The measured kinetic energy of the photoelectron differs therefore from its initial kinetic energy (when released in vacuum) by the potential difference between the sample and the detector:
Ekin,f = Ekin,i+ ∆V = hν − EB− φs+ ∆V = hν − EB− φd
Therefore the difference of the work functions has to be taken into account if one wants to determine the binding energy of the electron [17]. To determine the detector work function of the detector φd the Fermi level (EB = 0) needs to
be determined exactly. A polycrystalline gold reference is used since it exhibits a clear Fermi step edge.
3.2.3 Angle resolved photoelectron spectroscopy (ARPES)
ARPES is a method to measure both the kinetic energy and the momentum of the photoelectron. Additionally to normal PES, one can resolve the emis-sion angle, that can be related to the momentum of the photoelectron. The momentum of the initial N-particle system is changed by the momentum of the photon. The photon momentum (khν = 2πλ) is negligible compared to the
electron momentum (ke=¯h1p2m/E).
kNi − kNf = khν = 0
After being ejected from the sample surface, the momentum vector components of the photoelectrons can be generally expressed as:
Kx=
1 ¯ h
p
2mEkinsin θ cos φ
Ky =
1 ¯ h
p
2mEkinsin θ sin φ
Kz=
1 ¯ h
p
2mEkincos θ
In this expression, θ and φ are the emission angles relative to the surface normal. It is instructive to take a closer look at the wave vectors inside the sample. The potential of an infinite crystal (U (r)) is invariant after a translation by the typical unit cell dimension (R): U (r) = U (r + R). The wave vectors of
the electrons in such a crystal are translationally invariant as well, and they can be described by Bloch waves Ψnk = eik·runk(r), where unk(r + R) =
unk(r) is periodic [18]. As a consequence of eiG·R = 1, the momentum vector
is translation invariant in reciprocal space by G. The wave vector k0 is said to be in the nth Brillouin zone (BZ) when k0 = k + nG. Momenta inside the first BZ can be observed by measuring photoelectrons that are ejected about the normal emission angle. Higher order BZs can eventually be observed by measuring higher emission angles away from the surface normal.
Due to translational symmetry along the surface, the parallel wave vector component of the emitted photoelectrons is conserved when photoelectrons are ejected from the surface [19]. Thus the parallel component of the photoelectron momentum vector inside the solid (k||) is equal to the parallel component of the
momentum vector outside the solid (K||):
k||= K||=
1 ¯ h
p
2mEkinsin θ(cos φˆx + sin φ ˆy)
The last equation also sets the resolution of a photoemission experiment. | ∆k|||= ∆k||=
1 ¯ h
p
2mEkincos ∆θ (3.1)
As described above, one can probe the 2D projection of the 3D electronic ture by angle dependent measurements. A 3D mapping of the electronic struc-ture is less trivial to measure, however. Photoelectrons namely have to overcome a potential step that is caused by the potential difference between the crystal and the vacuum. Due to the potential step at the surface, the momentum perpendicular to the surface (k⊥) is not conserved and therefore it cannot be
determined directly. The precision of the perpendicular momentum component is relatively small as a result of the uncertainty principle and the small inelastic mean free path (IMFP, equation 3.8) of the photoelectrons [11]. There are dif-ferent methods to determine the perpendicular momentum vector, the method used for this project will now be explained. As a good approximation one can estimate the unbound photoelectrons to be ejected into a nearly free electron final state which is a parabola:
Ef(k) =
¯ h2(k2
⊥+ k2k)
2m − | E0| (3.2)
In the formula given above, Ef(k) represents the final state binding energy
(relative to EF) of the photoelectron in the bulk and E0 is the binding energy
that corresponds to the bottom of valence band. From equation (3.2) and the relations Ef = Ekin+ φ and
¯ h2k||2 2m = Ekin· sin 2θ it follows that [?]: k⊥= 1 ¯ h p
2m(Ekincos2(θ) + (E0+ φ)) (3.3)
Given the formula above and the fact that the kinetic energy of the pho-toelectron (Ekin) (relative to the vacuum energy) directly corresponds to the
photon energy (hν), the perpendicular momentum component k⊥can be probed
by the variation of photon energy (hν). The term (E0+ φ) = V0 is called the
inner potential, literature was consulted for this project to determine the inner potential [20].
Now that the conservation rules are introduced, I will describe the kinematics of the photoemission process. It is elucidative to describe the photoemission of an electron by the three-step-model, although one should keep in mind that quantum mechanically photoemission is more accurately described as a one-step process [19]. The three-one-step model simplifies the process into the following three steps. The first step is the optical excitation of the electron when it receives energy of the photon. Secondly, the electron propagating through the material to the sample surface. During the propagation it can collide with both electrons (electron-electron interactions) or ions (electron-phonon interactions) thereby losing energy. The electrons that did undergo a collision are called secondary electrons. At last, in the third step, the electrons with enough energy overcome the potential barrier and propagate into the vacuum. The advantage of the three step model is that physical events can be, as a good approximation, separately described.
The Fermi golden rule facilitates the calculation of the (optical excitation) transition probability from initial state (ΨN
i ) into final state (ΨNf ) in the
pres-ence of a continuous perturbation (Hint) [19]:
wf i= 2π ¯ h | hψ N f | Hint| ψiNi |2δ(EfN − EiN− hν)
In a photoemission process the transition of an electron from one state to the other is induced by the photon beam, where A is the electromagnetic vector potential of the photon beam and p represents the electronic momentum oper-ator. Only linear terms of A are taken into account since higher order terms are not significant in the UV-regime. The photon interaction is then described by the following Hamiltonian.
Hint=
e
2mc(A · p + p · A) = e mcA · p
Above, the divergence of the vector potential ∆ · A is neglected, since the vector potential for UV light approximately does not vary on the atomic scale. Then [p, A] = −i¯h∆A = 0, this simplification is called the dipole approximation. The optical transition can be more simplified by what is called the sudden ap-proximation. This approximates that no interactions of the photoelectron and the photohole are present because photoelectron energies are relatively high compared to the photohole energy. Since this approximation depends on the interaction of the photoelectron and photohole, it therefore depends on the ma-terial as well. For strongly correlated cuprate superconductors it is known that this approximation is already accurate at photon energies of hν = 20eV [19]. The sudden approximation allows the wave function of the photoelectron and its surrounding to be written separately and therefore disregards many-body interactions in the photoemission process. The initial (ΨN
i ) and final (ΨNf)
states, used for the Fermi golden rule, are written as a product of separate wave functions. The separate wave functions are those of the bound electron (φki), the photoelectron (φkf), the initial state (ψiN −1) and the final state (ψN −1f ) of the N − 1 particle system. The standard antisymmetric operator (B) anti-symmetrizes the product wave function to obey the Pauli exclusion principle [19]. ΨNi = Bφ k iΨ N −1 i , Ψ N f = Bφ k fΨ N −1 f (3.4) hΨN
f | Hint| ΨNi i = hφkf | Hint| φkiihΨN −1m | Ψ N −1
At this point, the approximated states are inserted in Fermi’s golden rule. I(k, Ekin) = 2π ¯ h X f,i | hψN f | Hint| ψiNi | 2 δ(EfN− EN i − hv) (3.6) =X f,i | Mk f,i| 2X m | cm.i| 2 δ(Ekin+ EmN −1− E N i − hv) (3.7)
An expression for the photoemission intensity I(k, Ekin) as function of
mo-mentum k and kinetic energy of the photoelectron Ekin is found. Above, an
abbreviated expression for the one-electron dipole matrix elements was used: Mf,ik = hφ k f | Hint| φkii, M (k, E) = X f,i hφkf | Hint | φkiiδ(E N f − E N i − hv)
It shows that a transition is allowed when initial and final states of the elec-tron overlap. These matrix elements depend on the geometry of the experimen-tal setup, on the orbiexperimen-tal origin of the electronic states and on the photon energy and polarization by which the material is illuminated. The other abbreviated term expresses the transition probability that the removal of a photoelectron causes the system to transit from the initial state to state m [19].
| cm,i|2=| hΨN −1m | Ψ N −1 i i
The system is non-interacting when a transition takes place from the initial state to a single final state. In that case the emission spectrum will give a single peak at that specific energy. However multiple final states may play a role leaving | cm,i |2 non zero, the system is then called interacting [19]. The
non-zero transitions of interacting systems are then observed as satellite peaks in the spectrum. The summation over all possible transition coefficients is called the one-particle spectral function:
A(k, E) =X
m
| cm,i|2δ(Ekin+ EmN −1− E N i − hv)
In the derived expression for the photoemission intensity it was not taken into account that only occupied states can be observed. The complete expression is obtained by multiplication with the Fermi-Dirac distribution function f (E).
f (E) = 1
1 + e(E−EF)/kBT
This simplifies the complete expression for the measurable photoemission current, which can be written as:
I(k, E) =| M (k, E) |2·A(k, E) · f (E)
In case of non-favorable matrix elements (M (k, E)), the experimental in-tensity of faint bands can be enhanced by taking the second derivative of the spectrum. In this study, the second derivative of the intensity with respect to k was taken for some spectra to sharpen the band contrast. The second derivative of a spectrum may introduce new - spurious - band features therefore it can only be used with care. The second derivative data will be explicitly indicated in the results section.
3.2.4 X-ray Photoemission Spectroscopy (XPS)
XPS was used in this project to identify the surface contamination. XPS is also based on the photoelectric effect, however the probing photon energy is in the X-ray regime. In the X-ray regime, k-sensitivity is lost due to the poor resolution (Eq. 3.1). Photon energies correspond to large binding energies of the core electrons that are strongly localized on the atomic sites. Since core levels are strongly localized in real space they are completely delocalized in reciprocal space. For the above reasons, high photon energy measurements focus on resolving the typical binding energy of the core electron orbitals rather then the momentum. The core level binding energies do not significantly change when a material condenses into a solid [21]. Moreover, the set of binding energies of core level electrons are unique per specific element. Core level spectroscopy (i.e. XPS) can be therefore used as a robust way to uniquely identify the chemical composition of surface compounds.
Spin-orbit interaction, a relativistic effect, is fairly significant in core level spectroscopy when the orbit angular momentum becomes greater then 0. The total angular momentum j relates to the angular momentum l and spin s as follows j = l +s or j = l −s. When the orbital angular momentum is 1, the total angular momentum can be 1/2 or 3/2 resulting in two different energy levels [21]. The energy difference between the two total angular momenta causes the core level XPS signal peaks to split. The typical energy level splitting, induced by the spin-orbit interaction, is of the order of 0eV-5eV. Core level spectra are identified by comparison with reference spectra from literature [22]. The reference spectra include the spin-orbit effect. The core level reference data used were taken with an identical photon source as the source used for this project: Al Kα with a typical photon energy centered at 1486.6eV.
Not all spectral peaks directly correspond to the core level energies listed in the reference, but can be explained by the interactions in the material that follow upon the electron-hole excitations. One of those secondary effects was independently explained by Pierre Victor Auger in 1923 and Lise Meitner in 1922 [23]. When strongly bound electrons are excited and removed due to pho-toelectric effect, a hole is left behind in the bulk and a photoelectron is measured by the detector. A low binding energy electron can then reduce its energy by decaying to the hole energy level that was left behind by the photoelectron. The energy released by the decay can then excite another low binding energy electron (Auger electron) therefore ejecting it from the bulk. This second photoelectron is detected as well and its energy corresponds to the energy released by the pre-viously decayed electron minus the binding energy of the second photoelectron. This mechanism suggests that the kinetic energy of the Auger electron does not depend on the photon excitation energy. To summarize, the observed spectral peaks in an experiment of core-level spectroscopy correspond to the superposi-tion of the electronic binding energies and of the difference of binding energies due to the Auger effect.
The Auger transitions are labeled as follows. The first index labels the hole that was left behind by the photoelectron. The second index corresponds to the electron level from which the an electron decays, releasing its energy while decaying. The energy difference between these two levels is transfered to the Auger electron that is labeled by the third index. Thus the Auger electron depicted in figure 3.3(b) is labeled KL2/3L2/3 [22].
Figure 3.3: (a) The initial photoelectron process; a photoelectron leaves behind a hole. (b) This is fol-lowed by the decay of an elec-tron into the hole transferring energy to the Auger electron
Plasmon excitations can also induce additional peaks in the X-ray PES spec-trum. Plasmons are collective density oscillations of the free electron gas with quantized energies. When a photoelectron travels to the surface to be ejected it can excite these density oscillations. The coupling reduces the energy of the photoelectron by the plasmon energy. As a result, satellites in the spectrum will appear that correspond to the plasmon coupled photoelectrons. The satel-lites are peaks in the spectrum shifted by the plasmon energy relative to the uncoupled core photoelectron [21].
3.2.5 Inelastic mean free path (IMFP)
In the previous section the excitation process of the photoelectron was described. Once the photoelectrons are excited, they travel to the surface. During this process, some electrons may lose energy by inelastic scattering before they hit the detector [19]. The inelastically scattered electrons are called secondary electrons and add a background signal to the measured spectra. The IMFP (Fig. 3.4) of photoelectrons traveling towards the surface depends on electron-electron and electron-phonon interactions. Electron-electron interactions are dominant in most cases since electron-phonon interactions only dominate at low electron energies. The cross-section for electron-electron scattering mechanism is given by the following relation [11]:
dσ dΩdω = ¯ h2 (πea0)2 1 q2Im{− 1 ε(q, ω)}
The variables Ω and ω indicate the frequency and solid scattering angle, re-spectively. The cross-section depends strongly on the dielectric function (q, ω), which varies per material. However, at high kinetic energies (compared to bond energies) the electrons are, to a good approximation, described by the free-electron gas. The typical free-electronic energies in a photoemission experiment are
high enough to make this assumption. The dominant scattering mechanism then depends on the mean electron-electron distance rs which does not differ much
per material [11]. The IFMP (λ) is therefore comparable for most materials: λ−1'√3a0R Ekin rs3/2lnh 4 9π 2/3Ekin R r 2 s i (3.8) Where a0 is the Bohr radius (a0 = 0.528˚A), the R is the Rydberg constant
(R = 13.6eV). The fact that the IFMP is comparable for most material is illustrated by the universal curve (Fig. 3.4).
Fig. 3.4: The inelastic electron mean free path (IMFP), indicated for various materials, follows a universal curve [11].
In this project ARPES measurements were performed in the photon energy range hν = (20 − 130)eV. Core level spectroscopy was performed at photon energies ranging between hν = 150eV and hν = 1486.6eV. This matches with a typical electron mean free path between 3˚A and 11˚A. Therefore, thin overlayers of comparable thickness will mask the electronic fingerprints of the material. However, when the sample surface is properly cleaned and probed by ARPES, the short IMFP is an ideal property if the surface states are of prime interest.
3.2.6 Surface states
As described in the last subsection, the IMFP of the electrons causes ARPES to be extremely surface sensitive which is ideal when probing the surface states that arise from the nanowires that are situated on the germanium surface. How-ever, since the IMFP is larger than a single nanowire layer, it is expected that germanium states, from the bulk and the near-surface region, are probed by ARPES as well. For that reason we take a closer look at the surface related electronic features of a crystal. Bulk states are generally described via an infi-nite crystal model. Obviously, such a model fails near the surface, where a the crystal is truncated. This causes the introduction of states different from the bulk states that are confined to the surface region, namely surface states (Fig. 3.5).
Fig. 3.5: 1D crystal at the crystal surface interface depicting the crystal potential (solid line) and the surface states (dashed line) [24].
Although ARPES is extremely surface sensitive, the observed ARPES spec-tra is often well described by bulk band calculations [11]. Unperturbed bulk states extend up to the topmost surface layer and are therefore observed by ARPES, since the probing depth is several surface layers. Since surface states can be found only in projected bulk gaps, the surface state contribution to the spectrum is relatively small compared to the bulk state contribution[11]. Still, it is relevant to take a closer look at the origin of the surface related states. A broad range of surface state computation techniques have been successfully used to describe surface states, these are however crystal specific studies. It is explanatory to describe the general existence of surface states by applying a simple 1D model. For this 1D surface state model, this section will focus on the tight-binding approximation, because it is ideal for describing semiconductors such as germanium. The electronic states of a bulk material are in this respect treated as a superposition of s-states ψson a chain of N sites with lattice
spac-ing a [25]. Such a state is a linear combination of unperturbed atomic orbitals (LCAO) and is written as follows:
Ψ(r) =
N
X
n=1
cnψs(r − an)
When this wave function is inserted in the Schr˝odinger equation the following relation results:
N
X
n=1
[cn(E − Es) − (V (r) − U (rn))ψs(rn)] = 0
Where, E is the band energy of the system, Esis the energy of the s-state, V (r)
represents the periodic potential and U (r) is the on-site spherical potential [25]. When considering only nearest neighbor interactions, this results in the following recursion relation for the LCAO coefficients:
In this simplified formulation, α is the on-site atomic energy shift and γ is the inter-atomic energy term also labeled bond energy. The ends of the 1D tight binding chain (the surfaces), are labeled by n = 1 and n = N . At both ends of the chain the on-site matrix elements of the bulk α differ from those on the surface α0, which leads to the following relation:
(E − E0+ α0)c1+ γc2= 0
,
γcN −1+ (E − Es+ α0)cN = 0
Under the condition | α0− α | /γ > 0, complex wave vectors are allowed. These vectors are related to the surface states whose energy differ from the ones of the tight-binding bulk. Complex wave vectors κ = π/a + iq introduce states that decay into the bulk, i.e. surface states. The surface state energy dispersion relation is:
Ess= E0− α + 2γ cosh(qssa)
It can be compared to the bulk tight-binding dispersion relation: E = E0− α + 2γ cos(κa)
One observes that the surface state energy (Ess) exceeds the bulk state energy
(E) because cosh(qssa) > cos(0) = 1 and therefore one can conclude that the
surface states resulting from the tight binding model lie inside the bulk state band gap, as mentioned before [25]. These surface states, that differ from the bulk states due to the change in potential at the surface, are called Tamm states [11].
3.2.7 FAMoS
ARPES in this project were performed with two different experimental configu-rations; the lab-based configuration called FOM Amsterdam Momentum Space (FAMoS) microscope in Amsterdam and the One-squared (12) end station at
the BESSY-II synchrotron in Berlin.
The FAMoS and its components will be discussed first. The photoemission spectroscopy toolbox of the FAMoS lab-based configuration consists of: a UV Helium gas discharge lamp and an Aluminum X-ray monochromatic light source in combination with a hemispherical detector. The UV source radiation is pro-duced by the electron decay between two energy levels of Helium, namely from 1s2p to the lowest available state 1s. A Gammadata vacuum UV (VUV) source was used as the FAMoS light source and works, in a nutshell as follows. Helium is ionized by a spark plug, hereafter electrons are excited by a magnetic and electric field. The static magnetic field induces a steady circular current that causes a centripetal force. That force is balanced by a varying electric field at a resonance frequency that causes the electrons to be continuously excited and then decay. The dominant decay energy is 21.22eV, which is caused by 1s2p→1s1s decay and labeled HeIα decay. The beam is monochromized and focused to a 3mm diameter spot by a toroidal diffraction grating.
For XPS a Scienta SAX-100 X-ray source in combination with a Scienta XM-780 monochromator was used. Inside the X-ray source a cathode filament emits
electrons that are accelerated by an electrical potential (15kV) towards an alu-minium anode. Highly energetic electrons bombard the anode thereby creating 1s holes that are filled by electrons from the 2p shell. This typical decay step, from 2p to 1s, is called Kαemission. The source emits also at other, unfavorable
wavelengths due to other energy level transitions and bremsstrahlung [26]. A set of Bragg mirrors monochromizes and focuses the beam up to a 1mm diameter spot. The resulting linewidth of the X-ray spectrum was measured to be full width half maximum < 300meV around the peak spectrum of 1486.6eV. At the lab-based set-up, a Scienta SES-2002 hemispherical analyzer was used to measure the photoelectron energy and momentum. Photoelectrons are first accelerated, retarded and focused by a electrostatic lenses. Accelerating or re-tarding of the photoelectrons allows to scan a branch of kinetic energies. The angle of emission, that corresponds to the in-plane momentum, is selected by the analyzer slit. Once past through the slit, the photoelectrons paths are bent by an electrostatic potential between two concentric hemispheres. The potential difference causes the high kinetic energy electrons to follow a wider trajectory than those with lower energies. Therefore the kinetic energies of the electrons are selected by the potential difference of the hemisphere. As a consequence, the potential difference, the width of the slit and the acceptance angle of the lens determine the energy resolution. At the end of their hemispherical trajectory, the electrons are collected by an electron multiplier and the resulting electron bunches produce an image on a fluorescent screen. The fluorescent screen’s image is captured by a charge coupled device (CCD) detector. The electrons observed by the CCD detector vary by kinetic energy in one direction and vary in momentum in the other detector direction (Fig. 3.6). The detector therefore directly measures the beam intensity with kinetic energy varying perpendicular to the slit direction and momentum on varying along the slit direction.
Fig. 3.6: (a) Schematic image of a hemispherical analyzer [19]. (b) Side view of the FAMoS. The pink arrow indicates the path of the photons and the light blue arrows follow the photoelectron trajectory [27]. (c) An image of the manipulator, to which the sample holder can be attached, with the rotation degree of freedom indicated.
Both light sources (X-ray source, UV lamp) can be used in combination with the same detector. This is because the photon momentum can be
ne-glected, therefore the photon angle of incidence (from two different directions) is not important. The samples can be precisely rotated and translated by a UHV manipulator to align the samples with the detector. The FAMoS manipulator has four degrees of freedom corresponding to translation in x-y-z-directions and a rotational degree of freedom around the z-axis. Moreover, the manipulator can cool the samples down to a temperature of T=20K by a continuous flow of liquid helium.
The hardware modification of FAMoS allowed samples to be measured imme-diately after growth, all within UHV conditions. This capability is a major ad-vantage of the FAMoS. However, FAMoS is just like every laboratory ARPES configuration, limited by the energy range of the light sources. As stated in the ARPES theory section, the photoemission intensity depends on the photon en-ergy. The photoelectron current intensity may thus be optimized by fine tuning the photon energy to a specific spectrum. For ARPES a He light source was used with a typical photon energy of 21.22eV and a secondary emission line at 40.80eV. No other UV excitation energies are possible with a conventional He-based laboratory light source. Core levels were investigated with a Al KαX-ray
source, delivering monochromatic photons with typical energies of 1486.6eV. Because of these light source limitation, additional ARPES measurements were carried out at the BESSY-II synchrotron light source.
3.2.8 BESSY-II storage ring, One-Squared end station
A significant part of the ARPES measurements presented in this thesis were performed at the synchrotron BESSY-II. A synchrotron is a large particle ac-celerator that, in this case, is specifically designed as a light source covering a very broad spectral range, from the infrared to the hard X-ray. The work-ing principle behind synchrotron radiation is that accelerated charge, by the relativity principle, radiates [28]. In a synchrotron, electrons are accelerated and injected into a storage ring were a circular motion is maintained by mag-nets (Fig. 3.7(a)). The circular electron current decays over time because of electron-electron or electron-gas collisions, therefore the injection procedure is repeated to maintain a steady current [29]. Electrons traveling at velocities close to the speed of light emit at a broad spectrum centered at the critical frequency (ωc):
ωc=
3c 2ργ
3
. Where c is the speed of light and ρ represents the radius of the orbit. A modern synchrotron radius is typically 50m. The relativistic term γ is equal to (1 − (vc)2)−1/2 [28]. Radiation from the circular motion alone is not intense
enough and corresponds to a rather broad energy spectrum. Therefore the beam is monochromized and the beam flux is increased and tuned by undulators that are situated at specific output locations (end stations) in the storage ring. The end stations are located at the end of the beam where the experiments take place. Undulators induce an undulating motion to the electron beam by means of a periodic magnetic field (Fig. 3.7(b)). The periodicity causes the emitted spectrum to be nearly monochromatic due to constructive interference at each wiggle [29]. An undulator creates harmonic waves according to the following
relation. λn= λu 2γ2n(1 + K2 2 + γ 2θ2)
Above, θ is the emission angle. The wiggler parameter (K) is proportional to the magnetic field of the undulator Bmax, therefore the wavelength can be
tuned for each undulator. The spectral linewidth is expressed as ∆λ λ =
2n N, with
the number of periods N and harmonic index n [29]. Since the intensity I of the radiation is proportional to the square of the number of periods, the beam intensity of an undulator is much higher. Typical undulators have N ' 100 whereas N = 1 for normal bending magnets. It is clear that the intensity of the undulator is four orders of magnitude larger, while the linewidth much narrower. Outside the ring, radiation is guided, monochromized and focused to each specific end station by grazing incidence Bragg diffraction gratings and mirrors. For this thesis work, we used the 12end-station which is coupled to the UE112-PGM beamline of BESSY-II. A high-flux photon energy range of 4eV-250eV with a linewidth of 1meV below hν = 100eV was available. The typical beam size was approximately 465 × 140µm2[30]. High resolution ARPES was possible by means of a Scienta R8000 analyzer. The above described features were beneficial for the investigation of the 1D surface systems of this thesis work. A small beam spot size allows one to circumvent large surface defects. The resolved electronic spectrum could be optimized by varying the photon energy.
Fig. 3.7: (a) Synchrotron schematics [31]. (b) Undulator schematics [28].
3.2.9 UHV Suitcase
During this project, the samples were prepared in the laboratory UHV prepara-tion chamber, where all the preparaprepara-tion elements were optimized to our specific needs. Hence UHV transportation of the samples from the lab-based setup to BESSY-II was unavoidable. Up to 10 Omicron sample plates can be trans-ported in a customized UHV suitcase (Fig. 3.8). UHV conditions in the interior of the suitcase are reached via the FAMoS setup using large turbomolecular pumps and cryopumps. The suitcase can maintain UHV (10−10mbar) condition
for 72 hours and still be portable due to a highly compact SAES NEXTorr D 100-5 pump. The pump consist of two pump mechanisms: a non-evaporable getter (NEG) in combination with ion pumping [32]. The NEG is the dominant pump since it absorbs the majority of gases like O2, H2, CO and N2 [33]. Gas
molecules absorb onto the porous getter where they diffuse into the bulk. Since the NEG working principle is absorption, non reactive gases cannot be pumped. The NEG is therefore complemented by an ion pump. An ion pump ionizes the gas molecules before accelerating the ions to an electrode. The ions penetrate into the reactive titanium electrode. Periodically, both pumps can saturate, therefore regeneration by heating up to 450◦C is required to maintain the pump function.
Fig. 3.8: The left side of the photo shows the suitcase cross with six ends to which the red NEXT-Torr pump and light-grey pressure gauge are attached. The pump and gauge controllers are situated on the right side.
3.2.10 Low energy electron diffraction (LEED)
LEED is a surface science technique based on the interference pattern of elec-trons. The technique, in its early days (1928), was used by Davisson and Germer as experimental evidence for particle wave duality [34]. This wave-like behavior forms the basis of the interference pattern to be studied. A typical LEED setup allows the selection of the electron energy between 10eV and 200eV. The energy spectrum of the emitted electrons is broadened (FWHM of '0.1eV) because the electrons are thermally ejected from a cathode filament. The roughly monochro-matic electrons are accelerated towards the sample, forming a monochromonochro-matic plane wave propagating perpendicular to the sample’s surface. The incoming plane waves are Bragg diffracted by the crystal surface and thereby enhanced or attenuated, depending on the electron scattering direction. Elastic scattering is caused by the ion-core potential which is approximately spherical at short range. An electrostatic potential allows only the back-scattered electrons that lost no more than 1eV to reach the detector, thus only elastically scattered electrons are observed. Like ARPES the probing depth is determined by the inelastic mean free path. After diffraction, elastically scattered electrons arrive on a fluorescent screen. Intensely bombarded spots on the screen illuminate more than others. A CCD camera records the projected electronic diffraction pattern on the fluorescent screen. During this project a Specs ER-LEED de-vice allowed an electron kinetic energy range between 20eV and 1000eV which translates to 2.7˚A and 0.4˚A using the de Broglie equation (λ = h/√2meEkin),
