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The subject of this thesis is the fabrication and characterization of magnetic topological
insulator layers of (V,Bi,Sb)\(_2\)Te\(_3\) exhibiting the quantum anomalous Hall
effect. A major task was the experimental realization of the quantum anomalous
Hall effect, which is only observed in layers with very specific structural,
electronic and magnetic properties. These properties and their influence on the
quantum anomalous Hall effect are analyzed in detail.
First, the optimal conditions for the growth of pure Bi\(_2\)Te\(_3\) and Sb\(_2\)Te\(_3\) crystal
layers and the resulting structural quality are studied. The crystalline quality of
Bi\(_2\)Te\(_3\) improves significantly at higher growth temperatures resulting in a small
mosaicity-tilt and reduced twinning defects. The optimal growth temperature is
determined as 260\(^{\circ}\)C, low enough to avoid desorption while maintaining a high
crystalline quality.
The crystalline quality of Sb\(_2\)Te\(_3\) is less dependent on the growth temperature.
Temperatures below 230\(^{\circ}\)C are necessary to avoid significant material desorption,
though. Especially for the nucleation on Si(111)-H, a low sticking coefficient is
observed preventing the coalescence of islands into a homogeneous layer.
The influence of the substrate type, miscut and annealing sequence on the growth
of Bi\(_2\)Te\(_3\) layers is investigated. The alignment of the layer changes depending on
the miscut angle and annealing sequence: Typically, layer planes align parallel to
the Si(111) planes. This can enhance the twin suppression due to transfer of the
stacking order from the substrate to the layer at step edges, but results in a step
bunched layer morphology. For specific substrate preparations, however, the layer
planes are observed to align parallel to the surface plane. This alignment avoids
displacement at the step edges, which would cause anti-phase domains. This results
in narrow Bragg peaks in XRD rocking curve scans due to long-range order in
the absence of anti-phase domains. Furthermore, the use of rough Fe:InP(111):B
substrates leads to a strong reduction of twinning defects and a significantly reduced
mosaicity-twist due to the smaller lattice mismatch.
Next, the magnetically doped mixed compound V\(_z\)(Bi\(_{1−x}\)Sb\(_x\))\(_{2−z}\)Te\(_3\) is studied in
order to realize the quantum anomalous Hall effect. The addition of V and Bi to
Sb\(_2\)Te\(_3\) leads to efficient nucleation on the Si(111)-H surface and a closed, homogeneous
layer. Magneto-transport measurements of layers reveal a finite anomalous
Hall resistivity significantly below the von Klitzing constant. The observation of
the quantum anomalous Hall effect requires the complete suppression of parasitic
bulklike conduction due to defect induced carriers. This can be achieved by optimizing
the thickness, composition and growth conditions of the layers.
The growth temperature is observed to strongly influence the structural quality.
Elevated temperatures result in bigger islands, improved crystallographic orientation
and reduced twinning. On the other hand, desorption of primarily Sb is
observed, affecting the thickness, composition and reproducibility of the layers.
At 190\(^{\circ}\)C, desorption is avoided enabling precise control of layer thickness and
composition of the quaternary compound while maintaining a high structural
quality.
It is especially important to optimize the Bi/Sb ratio in the (V,Bi,Sb)\(_2\)Te\(_3\) layers,
since by alloying n-type Bi\(_2\)Te\(_3\) and p-type Sb\(_2\)Te\(_3\) charge neutrality is achieved at
a specific mixing ratio. This is necessary to shift the Fermi level into the magnetic
exchange gap and fully suppress the bulk conduction. The Sb content x furthermore
influences the in-plane lattice constant a significantly. This is utilized to
accurately determine x even for thin films below 10 nm thickness required for the
quantum anomalous Hall effect. Furthermore, x strongly influences the surface
morphology: with increasing x the island size decreases and the RMS roughness
increases by up to a factor of 4 between x = 0 and x = 1.
A series of samples with x varied between 0.56-0.95 is grown, while carefully
maintaining a constant thickness of 9 nm and a doping concentration of 2 at.% V.
Magneto-transport measurements reveal the charge neutral point around x = 0.86
at 4.2 K. The maximum of the anomalous Hall resistivity of 0.44 h/e\(^2\) is observed
at x = 0.77 close to charge neutrality. Reducing the measurement temperature
to 50 mK significantly increases the anomalous Hall resistivity. Several samples
in a narrow range of x between 0.76-0.79 show the quantum anomalous Hall effect
with the Hall resistivity reaching the von Klitzing constant and a vanishing
longitudinal resistivity. Having realized the quantum anomalous Hall effect as the
first group in Europe, this breakthrough enabled us to study the electronic and
magnetic properties of the samples in close collaborations with other groups.
In collaboration with the Physikalisch-Technische Bundesanstalt high-precision
measurements were conducted with detailed error analysis yielding a relative de-
viation from the von Klitzing constant of (0.17 \(\pm\) 0.25) * 10\(^{−6}\). This is published
as the smallest, most precise value at that time, proving the high quality of the
provided samples. This result paves the way for the application of magnetic topological
insulators as zero-field resistance standards.
Non-local magneto-transport measurements were conducted at 15 mK in close
collaboration with the transport group in EP3. The results prove that transport
happens through chiral edge channels. The detailed analysis of small anomalies in
transport measurements reveals instabilities in the magnetic phase even at 15 mK.
Their time dependent nature indicates the presence of superparamagnetic contributions
in the nominally ferromagnetic phase.
Next, the influence of the capping layer and the substrate type on structural properties
and the impact on the quantum anomalous Hall effect is investigated. To
this end, a layer was grown on a semi-insulating Fe:InP(111)B substrate using the
previously optimized growth conditions. The crystalline quality is improved significantly
with the mosaicity twist reduced from 5.4\(^{\circ}\) to 1.0\(^{\circ}\). Furthermore, a layer
without protective capping layer was grown on Si and studied after providing sufficient
time for degradation. The uncapped layer on Si shows perfect quantization,
while the layer on InP deviates by about 5%. This may be caused by the higher
crystalline quality, but variations in e.g. Sb content cannot be ruled out as the
cause. Overall, the quantum anomalous Hall effect seems robust against changes
in substrate and capping layer with only little deviations.
Furthermore, the dependence of the quantum anomalous Hall effect on the thickness
of the layers is investigated. Between 5-8 nm thickness the material typically
transitions from a 2D topological insulator with hybridized top and bottom surface
states to a 3D topological insulator. A set of samples with 6 nm, 8 nm, and
9 nm thickness exhibits the quantum anomalous Hall effect, while 5 nm and 15 nm
thick layers show significant bulk contributions. The analysis of the longitudinal
and Hall conductivity during the reversal of magnetization reveals distinct differences
between different thicknesses. The 6 nm thick layer shows scaling consistent
with the integer quantum Hall effect, while the 9 nm thick layer shows scaling expected
for the topological surface states of a 3D topological insulator. The unique
scaling of the 9 nm thick layer is of particular interest as it may be a result of
axion electrodynamics in a 3D topological insulator.
Subsequently, the influence of V doping on the structural and magnetic properties
of the host material is studied systematically. Similarly to Bi alloying, increased
V doping seems to flatten the layer surface significantly. With increasing V content,
Te bonding partners are observed to increase simultaneously in a 2:3 ratio
as expected for V incorporation on group-V sites. The linear contraction of the
in-plane and out-of-plane lattice constants with increasing V doping is quantitatively
consistent with the incorporation of V\(^{3+}\) ions, possibly mixed with V\(^{4+}\)
ions, at the group-V sites. This is consistent with SQUID measurements showing
a magnetization of 1.3 \(\mu_B\) per V ion.
Finally, magnetically doped topological insulator heterostructures are fabricated
and studied in magneto-transport. Trilayer heterostructures with a non-magnetic
(Bi,Sb)\(_2\)Te\(_3\) layer sandwiched between two magnetically doped layers are predicted
to host the axion insulator state if the two magnetic layers are decoupled and in
antiparallel configuration. Magneto-transport measurements of such a trilayer heterostructure
with 7 nm undoped (Bi,Sb)\(_2\)Te\(_3\) between 2 nm thick layers doped with
1.5 at.% V exhibit a zero Hall plateau representing an insulating state. Similar results
in the literature were interpreted as axion insulator state, but in the absence
of a measurement showing the antiparallel magnetic orientation other explanations
for the insulating state cannot be ruled out.
Furthermore, heterostructures including a 2 nm thin, highly V doped layer region
show an anomalous Hall effect of opposite sign compared to previous samples. A
dependency on the thickness and position of the doped layer region is observed,
which indicates that scattering at the interfaces causes contributions to the anomalous
Hall effect of opposite sign compared to bulk scattering effects.
Many interesting phenomena in quantum anomalous Hall insulators as well as axion
insulators are still not unambiguously observed. This includes Majorana bound
states in quantum anomalous Hall insulator/superconductor hybrid systems and
the topological magneto-electric effect in axion insulators. The limited observation
temperature of the quantum anomalous Hall effect of below 1 K could be increased
in 3D topological insulator/magnetic insulator heterostructures which utilize the
magnetic proximity effect.
The main achievement of this thesis is the reproducible growth and characterization
of (V,Bi,Sb)2Te3 layers exhibiting the quantum anomalous Hall effect. The
detailed study of the structural requirements of the quantum anomalous Hall effect
and the observation of the unique axionic scaling behavior in 3D magnetic
topological insulator layers leads to a better understanding of the nature of this
new quantum state. The high-precision measurements of the quantum anomalous
Hall effect reporting the smallest deviation from the von Klitzing constant
are an important step towards the realization of a zero-field quantum resistance
standard.
Quantencomputer können manche Probleme deutlich effizienter lösen als klassische Rechner. Bisherige Umsetzungen leiden jedoch an einer zu geringen Dekohärenzzeit, weshalb die Lebenszeit der Quantenzustände einen limitierenden Faktor darstellt. Topologisch geschützte Anregungen, wie Majorana-Fermionen, könnten hingegen dieses Hindernis überwinden. Diese lassen sich beispielsweise in topologischen Supraleitern realisieren. Bis zum jetzigen Zeitpunkt existieren nur wenige Materialien, die dieses Phänomen aufweisen. Daher ist das Verständnis der elektronischen Eigenschaften für solche Verbindungen von großer Bedeutung.
In dieser Dissertation wird die Koexistenz von Supraleitung an der Probenoberfläche und topologischem Oberflächenzustand (engl. topological surface state, TSS) auf potentiellen topologischen Supraleitern überprüft. Diese beiden Bedingungen sind essentiell zur Ausbildung von topologischer Supraleitung in zeitumkehrgeschützten Systemen. Hierzu wird mittels Landaulevelspektroskopie und Quasiteilcheninterferenz das Vorhandensein des TSS am Ferminiveau auf Tl$_{x}$Bi$_{2}$Te$_{3}$ und Nb$_{x}$Bi$_{2}$Se$_{3}$ verifiziert, die mittels Transportmessungen als supraleitend identifiziert wurden. Anschließend folgen hochaufgelöste Spektroskopien an der Fermienergie, um die supraleitenden Eigenschaften zu analysieren.
Zur Interpretation der analysierten Eigenschaften wird zu Beginn der Ni-haltige Schwere-Fermion-Supraleiter TlNi$_{2}$Se$_{2}$ untersucht, der eine vergleichbare Übergangstemperatur besitzt. Anhand diesem werden die gängigen Messmethoden der Rastertunnelmikroskopie und -spektroskopie für supraleitende Proben vorgestellt und die Leistungsfähigkeit der Messapparatur demonstriert. Im Einklang mit der Literatur zeigt sich ein $s$-Wellencharakter des Paarungsmechanismus sowie die Formation eines für Typ~II-Supraleiter typischen Abrikosov-Gitters in schwachen externen Magnetfeldern.
Im folgenden Teil werden die potentiellen topologischen Supraleiter Tl$_{x}$Bi$_{2}$Te$_{3}$ und Nb$_{x}$Bi$_{2}$Se$_{3}$ begutachtet, für die eindeutig ein TSS bestätigt wird. Allerdings weisen beide Materialien keine Oberflächensupraleitung auf, was vermutlich durch eine Entkopplung der Oberfläche vom Volumen durch Bandverbiegung zu erklären ist. Unbeabsichtigte Kollisionen der Spitze mit der Probe führen jedoch zu supraleitenden Spitzen, die wesentlich erhöhte Werte für die kritische Temperatur und das kritische Feld zeigen.
Der letzte Abschnitt widmet sich dem supraleitenden Substrat Nb(110), für den der Reinigungsprozess erläutert wird. Hierbei sind kurze Heizschritte bis nahe des Schmelzpunktes nötig, um die bei Umgebungsbedingungen entstehende Sauerstoffrekonstruktion effektiv zu entfernen. Des Weiteren werden die elektronischen Eigenschaften untersucht, die eine Oberflächenresonanz zum Vorschein bringen. Hochaufgelöste Messungen lassen eine durch die BCS-Theorie gut repräsentierte Struktur der supraleitenden Energielücke erkennen. Magnetfeldabhängige Experimente offenbaren zudem eine mit der Kristallstruktur vereinbare Anisotropie des Paarungspotentials. Mit diesen Erkenntnissen kann Nb(110) zukünftig als Ausgang für das Wachstum von topologischen Supraleitern herangezogen werden.
This thesis describes the studies of topological superconductivity, which is predicted to
emerge when pair correlations are induced into the surface states of 2D and 3D topolog-
ical insulators (TIs). In this regard, experiments have been designed to investigate the
theoretical ideas first pioneered by Fu and Kane that in such system Majorana bound
states occur at vortices or edges of the system [Phys. Rev. Lett. 100, 096407 (2008), Phys.
Rev. B 79, 161408 (2009)]. These states are of great interest as they constitute a new
quasiparticle which is its own antiparticle and can be used as building blocks for fault
tolerant topological quantum computing.
After an introduction in chapter 1, chapter 2 of the thesis lays the foundation for the
understanding of the field of topology in the context of condensed matter physics with a
focus on topological band insulators and topological superconductors. Starting from a
Chern insulator, the concepts of topological band theory and the bulk boundary corre-
spondence are explained. It is then shown that the low energy Hamiltonian of mercury
telluride (HgTe) quantum wells of an appropriate thickness can be written as two time
reversal symmetric copies of a Chern insulator. This leads to the quantum spin Hall effect.
In such a system, spin-polarized one dimensional conducting states form at the edges
of the material, while the bulk is insulating. This concept is extended to 3D topological
insulators with conducting 2D surface states. As a preliminary step to treating topological
superconductivity, a short review of the microscopic theory of superconductivity, i.e. the
theory of Bardeen, Cooper, and Shrieffer (BCS theory) is presented. The presence of
Majorana end modes in a one dimensional superconducting chain is explained using the
Kitaev model. Finally, topological band insulators and conventional superconductivity
are combined to effectively engineer p-wave superconductivity. One way to investigate
these states is by measuring the periodicity of the phase of the Josephson supercurrent
in a topological Josephson junction. The signature is a 4π-periodicity compared to the
2π-periodicity in conventional Josephson junctions. The proof of the presence of this
effect in HgTe based Josephson junction is the main goal of this thesis and is discussed in
chapters 3 to 6.
Chapter 3 describes in detail the transport of a 3D topological insulator based weak
link under radio-frequency radiation. The chapter starts with a review of the state of
research of (i) strained HgTe as 3D topological insulator and (ii) the progress of induc-
ing superconducting correlations into the topological surface states and the theoretical
predictions of 3D TI based Josephson junctions. Josephson junctions based on strained
HgTe are successfully fabricated. Before studying the ac driven Josephson junctions, the
dc transport of the devices is analysed. The critical current as a function of temperature
is measured and it is possible to determine the induced superconducting gap. Under
rf illumination Shapiro steps form in the current voltage characteristic. A missing first
step at low frequencies and low powers is found in our devices. This is a signature of
a 4π-periodic supercurrent. By studying the device in a wide parameter range - as a
147148 SUMMARY
function of frequency, power, device geometry and magnetic field - it is shown that the
results are in agreement with the presence of a single gapless Andreev doublet and several
conventional modes.
Chapter 4 gives results of the numerical modelling of the I −V dynamics in a Josephson
junction where both a 2π- and a 4π-periodic supercurrents are present. This is done in
the framework of an equivalent circuit representation, namely the resistively shunted
Josephson junction model (RSJ-model). The numerical modelling is in agreement with
the experimental results in chapter 3. First, the missing of odd Shapiro steps can be
understood by a small 4π-periodic supercurrent contribution and a large number of
modes which have a conventional 2π-periodicity. Second, the missing of odd Shapiro
steps occurs at low frequency and low rf power. Third, it is shown that stochastic processes
like Landau Zener tunnelling are most probably not responsible for the 4π contribution.
In a next step the periodicity of Josephson junctions based on quantum spin Hall
insulators using are investigated in chapter 5. A fabrication process of Josephson junctions
based on inverted HgTe quantum wells was successfully developed. In order to achieve a
good proximity effect the barrier material was removed and the superconductor deposited
without exposing the structure to air. In a next step a gate electrode was fabricated which
allows the chemical potential of the quantum well to be tuned. The measurement of the
diffraction pattern of the critical current Ic due to a magnetic field applied perpendicular
to the sample plane was conducted. In the vicinity to the expected quantum spin Hall
phase, the pattern resembles that of a superconducting quantum interference device
(SQUID). This shows that the current flows predominantly on the edges of the mesa.
This observation is taken as a proof of the presence of edge currents. By irradiating the
sample with rf, missing odd Shapiro steps up to step index n = 9 have been observed. This
evidences the presence of a 4π-periodic contribution to the supercurrent. The experiment
is repeated using a weak link based on a non-inverted HgTe quantum well. This material
is expected to be a normal band insulator without helical edge channels. In this device,
all the expected Shapiro steps are observed even at low frequencies and over the whole
gate voltage range. This shows that the observed phenomena are directly connected
to the topological band structure. Both features, namely the missing of odd Shapiro
steps and the SQUID like diffraction pattern, appear strongest towards the quantum spin
Hall regime, and thus provide evidence for induced topological superconductivity in the
helical edge states.
A more direct way to probe the periodicity of the Josephson supercurrent than using
Shapiro steps is the measurement of the emitted radiation of a weak link. This experiment
is presented in chapter 6. A conventional Josephson junction converts a dc bias V to
an ac current with a characteristic Josephson frequency fJ
= eV /h. In a topological
Josephson junction a frequency at half the Josephson frequency fJ /2 is expected. A
new measurement setup was developed in order to measure the emitted spectrum of a
single Josephson junction. With this setup the spectrum of a HgTe quantum well based
Josephson junction was measured and the emission at half the Josephson frequency fJ /2
was detected. In addition, fJ emission is also detected depending on the gate voltage and
detection frequency. The spectrum is again dominated by half the Josephson emission at
low voltages while the conventional emission is determines the spectrum at high voltages.
A non-inverted quantum well shows only conventional emission over the whole gateSUMMARY 149
voltage and frequency range. The linewidth of the detected frequencies gives a measure
on the lifetime of the bound states: From there, a coherence time of 0.3–4ns for the fJ /2
line has been deduced. This is generally shorter than for the fJ line (3–4ns).
The last part of the thesis, chapter 7, reports on the induced superconducting state
in a strained HgTe layer investigated by point-contact Andreev reflection spectroscopy.
For the experiment, a HgTe mesa was fabricated with a small constriction. The diameter
of the orifice was chosen to be smaller than the mean free path estimated from magne-
totransport measurements. Thus one gets a ballistic point-contact which allows energy
resolved spectroscopy. One part of the mesa is covered with a superconductor which
induces superconducting correlations into the surface states of the topological insulator.
This experiment therefore probes a single superconductor normal interface. In contrast to
the Josephson junctions studied previously, the geometry allows the acquisition of energy
resolved information of the induced superconducting state through the measurement
of the differential conductance dI/dV as a function of applied dc bias for various gate
voltages, temperatures and magnetic fields. An induced superconducting order parame-
ter of about 70µeV was extracted but also signatures of the niobium gap at the expected
value around Δ Nb
≈ 1.1meV have been found. Simulations using the theory developed by
Blonder, Tinkham and Klapwijk and an extended model taking the topological surface
states into account were used to fit the data. The simulations are in agreement with a
small barrier at the topological insulator-induced topological superconductor interface
and a high barrier at the Nb to topological insulator interface. To understand the full con-
ductance curve as a function of applied voltage, a non-equilibrium driven transformation
is suggested. The induced superconductivity is suppressed at a certain bias value due to
local electron population. In accordance with this suppression, the relevant scattering
regions change spatially as a function of applied bias.
To conclude, it is emphasized that the experiments conducted in this thesis found
clear signatures of induced topological superconductivity in HgTe based quantum well
and bulk devices and opens up the avenue to many experiments. It would be interesting
to apply the developed concepts to other topological matter-superconductor hybrid
systems. The direct spectroscopy and manipulation of the Andreev bound states using
circuit quantum electrodynamic techniques should be the next steps for HgTe based
samples. This was already achieved in superconducting atomic break junctions by the
group in Saclay [Science 2015, 349, 1199-1202 (2015)]. Another possible development
would be the on-chip detection of the emitted spectrum as a function of the phase φ
through the junction. In this connection, the topological junction needs to be shunted
by a parallel ancillary junction. Such a setup would allow the current phase relation
I(φ) directly and the lifetime of the bound states to be measured directly. By coupling
this system to a spectrometer, which can be another Josephson junction, the energy
dependence of the Andreev bound states E(φ) could be obtained. The experiments on
the Andreev reflection spectroscopy described in this thesis could easily be extended to
two dimensional topological insulators and to more complex geometries, like a phase
bias loop or a tunable barrier at the point-contact. This work might also be useful for
answering the question how and why Majorana bound states can be localized in quantum
spin Hall systems.
Numerical Simulations of Heavy Fermion Systems: From He-3 Bilayers to Topological Kondo Insulators
(2014)
Even though heavy fermion systems have been studied for a long time, a strong interest in heavy fermions persists to this day. While the basic principles of local moment formation, Kondo effect and formation of composite quasiparticles leading to a Fermi liquid, are under- stood, there remain many interesting open questions. A number of issues arise due to the interplay of heavy fermion physics with other phenomena like magnetism and superconduc- tivity.
In this regard, experimental and theoretical investigations of He-3 can provide valuable insights. He-3 represents a unique realization of a quantum liquid. The fermionic nature of He-3 atoms, in conjunction with the absence of long-range Coulomb repulsion, makes this material an ideal model system to study Fermi liquid behavior.
Bulk He-3 has been investigated for quite some time. More recently, it became possible to prepare and study layered He-3 systems, in particular single layers and bilayers. The pos- sibility of tuning various physical properties of the system by changing the density of He-3 and using different substrate materials makes layers of He-3 an ideal quantum simulator for investigating two-dimensional Fermi liquid phenomenology.
In particular, bilayers of He-3 have recently been found to exhibit heavy fermion behavior. As a function of temperature, a crossover from an incoherent state with decoupled layers to a coherent Fermi liquid of composite quasiparticles was observed. This behavior has its roots in the hybridization of the two layers. The first is almost completely filled and subject to strong correlation effects, while the second layer is only partially filled and weakly correlated. The quasiparticles are formed due to the Kondo screening of localized moments in the first layer by the second-layer delocalized fermions, which takes place at a characteristic temperature scale, the coherence scale Tcoh.
Tcoh can be tuned by changing the He-3 density. In particular, at a certain critical filling,
the coherence scale is expected to vanish, corresponding to a divergence of the quasiparticle effective mass, and a breakdown of the Kondo effect at a quantum critical point. Beyond the critical point, the layers are decoupled. The first layer is a local moment magnet, while the second layer is an itinerant overlayer.
However, already at a filling smaller than the critical value, preempting the critical point, the onset of a finite sample magnetization was observed. The character of this intervening phase remained unclear.
Motivated by these experimental observations, in this thesis the results of model calcula- tions based on an extended Periodic Anderson Model are presented. The three particle ring exchange, which is the dominant magnetic exchange process in layered He-3, is included in the model. It leads to an effective ferromagnetic interaction between spins on neighboring sites. In addition, the model incorporates the constraint of no double occupancy by taking the limit of large local Coulomb repulsion.
By means of Cellular DMFT, the model is investigated for a range of values of the chemical potential µ and inverse temperature β = 1/T . The method is a cluster extension to the Dy- namical Mean-Field Theory (DMFT), and allows to systematically include non-local correla- tions beyond the DMFT. The auxiliary cluster model is solved by a hybridization expansion CTQMC cluster solver, which provides unbiased, numerically exact results for the Green’s function and other observables of interest.
As a first step, the onset of Fermi liquid coherence is studied. At low enough temperature, the self-energy is found to exhibit a linear dependence on Matsubara frequency. Meanwhile, the spin susceptibility crossed over from a Curie-Weiss law to a Pauli law. Both observations serve as fingerprints of the Fermi liquid state.
The heavy fermion state appears at a characteristic coherence scale Tcoh. This scale depends strongly on the density. While it is rather high for small filling, for larger filling Tcoh is increas- ingly suppressed. This involves a decreasing quasiparticle residue Z ∼ Tcoh and an enhanced mass renormalization m∗/m ∼ Tcoh−1. Extrapolation leads to a critical filling, where the co-
herence scale is expected to vanish at a quantum critical point. At the same time, the effective mass diverges. This corresponds to a breakdown of the Kondo effect, which is responsible for the formation of quasiparticles, due to a vanishing of the effective hybridization between the layers.
Taking only single-site DMFT results into account, the above scenario seems plausible. However, paramagnetic DMFT neglects the ring exchange interaction completely. In or- der to improve on this, Cellular DMFT simulations are conducted for small clusters of size Nc = 2 and 3. The results paint a different physical picture. The ring exchange, by favor- ing a ferromagnetic alignment of spins, competes with the Kondo screening. As a result, strong short-range ferromagnetic fluctuations appear at larger values of µ. By lowering the temperature, these fluctuations are enhanced at first. However, for T < Tcoh they are increas- ingly suppressed, which is consistent with Fermi liquid coherence. However, beyond a certain threshold value of µ, fluctuations persist to the lowest temperatures. At the same time, while not apparent in the DMFT results, the total occupation n increases quite strongly in a very narrow range around the same value of µ. The evolution of n with µ is always continuous, but hints at a discontinuity in the limit Nc → ∞. This first-order transition breaks the Kondo effect. Beyond the transition, a ferromagnetic state in the first layer is established, and the second layer becomes a decoupled overlayer.
These observations provide a quite appealing interpretation of the experimental results. As a function of chemical potential, the Kondo breakdown quantum critical point is preempted by a first-order transition, where the layers decouple and the first layer turns into a ferromagnet. In the experimental situation, where the filling can be tuned directly, the discontinuous transition is mirrored by a phase separation, which interpolates between the Fermi liquid ground state at lower filling and the magnetic state at higher filling. This is precisely the range of the intervening phase found in the experiments, which is characterized by an onset of a finite sample magnetization.
Besides the interplay of heavy fermion physics and magnetic exchange, recently the spin- orbit coupling, which is present in many heavy fermion materials, attracted a lot of interest. In the presence of time-reversal symmetry, due to spin-orbit coupling, there is the possibility of a topological ground state.
It was recently conjectured that the energy scale of spin-orbit coupling can become dom- inant in heavy fermion materials, since the coherence scale and quasiparticle bandwidth are rather small. This can lead to a heavy fermion ground state with a nontrivial band topology; that is, a topological Kondo insulator (TKI). While being subject to strong correlation effects, this state must be adiabatically connected to a non-interacting, topological state.
The idea of the topological ground state realized in prototypical Kondo insulators, in par- ticular SmB6, promises to shed light on some of the peculiarities of these materials, like a residual conductivity at the lowest temperatures, which have remained unresolved so far.
In this work, a simple two-band model for two-dimensional topological Kondo insulators is devised, which is based on a single Kramer’s doublet coupled to a single conduction band. The model is investigated in the presence of a Hubbard interaction as a function of interaction strength U and inverse temperature β. The bulk properties of the model are obtained by DMFT, with a hybridization expansion CTQMC impurity solver. The DMFT approximation of a local self-energy leads to a very simple way of computing the topological invariant.
The results show that with increasing U the system can be driven through a topological phase transition. Interestingly, the transition is between distinct topological insulating states, namely the Γ-phase and M-phase. This appearance of different topological phases is possible due to the symmetry of the underlying square lattice. By adiabatically connecting both in- teracting states with the respective non-interacting state, it is shown that the transition indeed drives the system from the Γ-phase to the M-phase.
A different behavior can be observed by pushing the bare position of the Kramer’s doublet to higher binding energies. In this case, the non-interacting starting point has a trivial band topology. By switching on the interaction, the system can be tuned through a quantum phase transition, with a closing of the band gap. Upon reopening of the band gap, the system is in the Γ-phase, i. e. a topological insulator. By increasing the interaction strength further, the system moves into a strongly correlated regime. In fact, close to the expected transition to the M phase, the mass renormalization becomes quite substantial. While absent in the para- magnetic DMFT simulations conducted, it is conceivable that instead of a topological phase transition, the system undergoes a time-reversal symmetry breaking, magnetic transition.
The regime of strong correlations is studied in more detail as a function of temperature, both in the bulk and with open boundary conditions. A quantity which proved very useful is the bulk topological invariant Ns, which can be generalized to finite interaction strength and temperature. In particular, it can be used to define a temperature scale T ∗ for the onset of the topological state. Rescaling the results for Ns, a nice data collapse of the results for different values of U, from the local moment regime to strongly mixed valence, is obtained. This hints at T ∗ being a universal low energy scale in topological Kondo insulators. Indeed, by comparing T ∗ with the coherence scale extracted from the self-energy mass renormalization, it is found that both scales are equivalent up to a constant prefactor. Hence, the scale T ∗ obtained from the temperature dependence of topological properties, can be used as an independent measure for Fermi liquid coherence. This is particularly useful in the experimentally relevant mixed valence regime, where charge fluctuations cannot be neglected. Here, a separation of the energy scales related to spin and charge fluctuations is not possible.
The importance of charge fluctuations becomes evident in the extent of spectral weight transfer as the temperature is lowered. For mixed valence, while the hybridization gap emerges, a substantial amount of spectral weight is shifted from the vicinity of the Fermi level to the lower Hubbard band. In contrast, this effect is strongly suppressed in the local moment regime.
In addition to the bulk properties, the spectral function for open boundaries is studied as a function of temperature, both in the local moment and mixed valence regime. This allows an investigation of the emergence of topological edge states with temperature. The method used here is the site-dependent DMFT, which is a generalization of the conventional DMFT to inhomogeneous systems. The hybridization expansion CTQMC algorithm is used as impurity solver.
By comparison with the bulk results for the topological quantity Ns, it is found that the
temperature scale for the appearance of the topological edge states is T ∗, both in the mixed valence and local moment regime.
Exploring the transport properties of the three-dimensional topological insulator material HgTe
(2015)
In the present thesis the transport properties of strained bulk HgTe devices are investigated. Strained HgTe forms a 3D TI and is of special interest for studying topological surface states, since it can be grown by MBE in high crystal quality. The low defect density leads to considerable mobility values, well above the mobilities of other TI materials. However, strained HgTe has a small band gap of ca. 20 meV. With respect to possible applications the question is important, under which conditions the surface transport occurs. To answer this question, the HgTe devices are investigated at dilution refrigerator temperatures (T<100 mK) in high magnetic fields of different orientation. The influence of top and back gate electrodes as well as surface protecting layers is discussed.
On the basis of an analysis of the quantum Hall behaviour it is shown that transport is dominated by the topological surface states in a surprisingly large parameter range. A dependence on the applied top gate voltage is presented for the topological surface states. It enables the first demonstration of an odd integer QHE sequence from the surfaces perpendicular to the magnetic field. Furthermore, the p-type QHE from the surface states is observed for the first time in any 3D TI. This is achieved in samples of high surface quality. It is concluded from the gate response that the screening behaviour in 3D TI devices is non-trivial. The transport data are qualitatively analysed by means of intuitive theoretical models.
After the discovery of three-dimensional topological insulators (TIs), such as tetradymite chalcogenides Bi$_2$Se$_3$, Bi$_2$Te$_3$ and Sb$_2$Te$_3$ – a new class of quantum materials characterized by their unique surface electronic properties – the solid state community got focused on topological states that are driven by strong electronic correlations and magnetism. An important material class is the magnetic TI (MTI) exhibiting the quantum anomalous Hall (QAH) effect, i.e. a dissipationless quantized edge-state transport in the absence of external magnetic field, originating from the interplay between ferromagnetism and a topologically non-trivial band structure. The unprecedented opportunities offered by these new exotic materials open a new avenue for the development of low-dissipation electronics, spintronics, and quantum computation. However, the major concern with QAH effect is its extremely low onset temperature, limiting its practical application. To resolve this problem, a comprehensive understanding of the microscopic origin of the underlying ferromagnetism is necessary.
V- and Cr-doped (Bi,Sb)$_2$Te$_3$ are the two prototypical systems that have been widely studied as realizations of the QAH state. Finding microscopic differences between the strongly correlated V and Cr impurities would help finding a relevant model of ferromagnetic coupling and eventually provide better control of the QAH effect in these systems. Therefore, this thesis first focuses on the V- and Cr-doped (Bi,Sb)$_2$Te$_3$ systems, to better understand these differences. Exploiting the unique capabilities of x-ray absorption spectroscopy and magnetic circular dichroism (XAS/XMCD), combined with advanced modeling based on multiplet ligand-field theory (MLFT), we provide a detailed microscopic insight into the local electronic and magnetic properties of these systems and determine microscopic parameters crucial for the comparison with theoretical models, which include the $d$-shell filling, spin and orbital magnetic moments. We find a strongly covalent ground state, dominated by the superposition of one and two Te-ligand-hole configurations, with a negligible contribution from a purely ionic 3+ configuration. Our findings indicate the importance of the Te $5p$ states for the ferromagnetism in (Bi, Sb)$_2$Te$_3$ and favor magnetic coupling mechanisms involving $pd$-exchange. Using state-of-the-art density functional theory (DFT) calculations in combination with XMCD and resonant photoelectron spectroscopy (resPES), we reveal the important role of the $3d$ impurity states in mediating magnetic exchange coupling. Our calculations illustrate that the kind and strength of the exchange coupling varies with the impurity $3d$-shell occupation. We find a weakening of ferromagnetic properties upon the increase of doping concentration, as well as with the substitution of Bi at the Sb site. Finally, we qualitatively describe the origin of the induced magnetic moments at the Te and Sb sites in the host lattice and discuss their role in mediating a robust ferromagnetism based on a $pd$-exchange interaction scenario. Our findings reveal important clues to designing higher $T_{\text{C}}$ MTIs.
Rare-earth ions typically exhibit larger magnetic moments than transition-metal ions and thus promise the opening of a wider exchange gap in the Dirac surface states of TIs, which is favorable for the realization of the high-temperature QAH effect. Therefore, we have further focused on Eu-doped Bi$_2$Te$_3$ and scrutinized whether the conditions for formation of a substantial gap in this system are present by combining spectroscopic and bulk characterization methods with theoretical calculations. For all studied Eu doping concentrations, our atomic multiplet analysis of the $M_{4,5}$ x-ray absorption and magnetic circular dichroism spectra reveals a Eu$^{2+}$ valence, unlike most other rare earth elements, and confirms a large magnetic moment. At temperatures below 10 K, bulk magnetometry indicates the onset of antiferromagnetic ordering. This is in good agreement with DFT results, which predict AFM interactions between the Eu impurities due to the direct overlap of the impurity wave functions. Our results support the notion of antiferromagnetism coexisting with topological surface states in rare-earth doped Bi$_2$Te$_3$ and corroborate the potential of such doping to result in an antiferromagnetic TI with exotic quantum properties.
The doping with impurities introduces disorder detrimental for the QAH effect, which may be avoided in stoichiometric, well-ordered magnetic compounds. In the last part of the thesis we have investigated the recently discovered intrinsic magnetic TI (IMTI) MnBi$_6$Te$_{10}$, where we have uncovered robust ferromagnetism with $T_{\text{C}} \approx 12$ K and connected its origin to the Mn/Bi intermixing. Our measurements reveal a magnetically intact surface with a large moment, and with FM properties similar to the bulk, which makes MnBi$_6$Te$_{10}$ a promising candidate for the QAH effect at elevated temperatures. Moreover, using an advanced ab initio MLFT approach we have determined the ground-state properties of Mn and revealed a predominant contribution of the $d^5$ configuration to the ground state, resulting in a $d$-shell electron occupation $n_d = 5.31$ and a large magnetic moment, in excellent agreement with our DFT calculations and the bulk magnetometry data. Our results together with first principle calculations based on the DFT-GGA$+U$, performed by our collaborators, suggest that carefully engineered intermixing plays a crucial role in achieving a robust long-range FM order and therefore could be the key for achieving enhanced QAH effect properties.
We expect our findings to aid better understanding of MTIs, which is essential to help increasing the temperature of the QAH effect, thus facilitating the realization of low-power electronics in the future.
A plethora of novel material concepts are currently being investigated in the condensed matter research community. Some of them hold promise to shape our everyday world in a way that silicon-based semiconductor materials and the related development of semiconductor devices have done in the past. In this regard, the last decades have witnessed an explosion of studies concerned with so called ‘’quantum materials’’ with emerging novel functionalities. These could eventually lead to new generations of electronic and/or spintronic devices. One particular material class, the so called topological materials, play a central role. As far as their technological applicability is concerned, however, they are still facing outstanding challenges to date.
Predicted for the first time in 2005 and experimentally verified in 2007, two-dimensional topological insulators (2D TIs) (a.k.a. quantum spin Hall insulators) exhibit the outstanding property of hosting spin-polarized metallic states along the boundaries of the insulating 2D bulk material, which are protected from elastic single-particle backscattering and give rise to the quantum spin Hall effect (QSHE). Owing to these peculiar properties the QSHE holds promise for dissipationless charge and/or spin transport. However, also in today’s best 2D TIs the observation of the QSHE is still limited to cryogenic temperatures of maximum 100 K. Here, the discovery of bismuthene on SiC(0001) has marked a milestone towards a possible realization of the QSHE at or beyond room-temperature owing to the massively increased electronic bulk energy gap on the order of 1 eV. This thesis is devoted to and motivated by the goal of advancing its synthesis and to build a deeper understanding of its one-particle and two-particle electronic properties that goes beyond prior work.
Regarding the aspect of material synthesis, an improved growth procedure for bismuthene is elaborated that increases the domain size of the material considerably (by a factor of ≈ 3.2 - 6.5 compared to prior work). The improved film quality is an important step towards any future device application of bismuthene, but also facilitates all further basic studies of this material.
Moreover, the deposition of magnetic transition metals (Mn and Co) on bismuthene is investigated. Thereby, the formation of ordered magnetic Bi-Mn/Co alloys is realized, their structure is resolved with scanning tunneling microscopy (STM), and their pristine electronic properties are resolved with scanning tunneling spectroscopy (STS) and photoemission spectroscopy (PES). It is proposed that these ordered magnetic Bi-Mn/Co-alloys offer the potential to study the interplay between magnetism and topology in bismuthene in the future.
In this thesis, a wide variety of spectroscopic techniques are employed that aim to build an understanding of the single-particle, as well as two-particle level of description of bismuthene's electronic structure. The techniques involve STS and angle-resolved PES (ARPES) on the one hand, but also optical spectroscopy and time-resolved ARPES (trARPES), on the other hand. Moreover, these experiments are accompanied by advanced numerical modelling in form of GW and Bethe-Salpeter equation calculations provided by our theoretical colleagues. Notably, by merging many experimental and theoretical techniques, this work sets a benchmark for electronic structure investigations of 2D materials in general.
Based on the STS studies, electronic quasi-particle interferences in quasi-1D line defects in bismuthene that are reminiscent of Fabry-Pérot states are discovered. It is shown that they point to a hybridization of two pairs of helical boundary modes across the line defect, which is accompanied by a (partial) lifting of their topological protection against elastic single-particle backscattering.
Optical spectroscopy is used to reveal bismuthene's two-particle elecronic structure. Despite its monolayer thickness, a strong optical (two-particle) response due to enhanced electron-hole Coulomb interactions is observed. The presented combined experimental and theoretical approach (including GW and Bethe-Salpeter equation calculations) allows to conclude that two prominent optical transitions can be associated with excitonic transitions derived from the Rashba-split valence bands of bismuthene. On a broader scope this discovery might promote further experiments to elucidate links of excitonic and topological physics.
Finally, the excited conduction band states of bismuthene are mapped in energy and momentum space employing trARPES on bismuthene for the first time. The direct and indirect band gaps are succesfully extracted and the effect of excited charge carrier induced gap-renormalization is observed. In addition, an exceptionally fast excited charge carrier relaxation is identified which is explained by the presence of a quasi-metallic density of states from coupled topological boundary states of domain boundaries.
Quantum point contacts (QPCs) are one-dimensional constrictions in an otherwise extended two-dimensional electron or hole system. Since their first realization in GaAs based two-dimensional electron gases, QPCs have become basic building blocks of mesoscopic physics and are used in manifold experimental contexts. A so far unrealized goal however is the implementation of QPCs in the new material class of two-dimensional topological insulators, which host the emergence of the so-called quantum spin Hall (QSH) effect. The latter is characterized by the formation of conducting one-dimensional spin-polarized states at the device edges, while the bulk is insulating. Consequently, an implemented QPC technology can be utilized to bring the QSH edge channels in close spatial proximity, thus for example enabling the study of interaction effects between the edge states. The thesis at hand describes the technological realization as well as the subsequent experimental characterization and analysis of QPCs in a QSH system for the first time.
After an introduction is given in Chapter 1, the subsequent Chapter 2 starts with discussing the peculiar band structure of HgTe. The emergence of the QSH phase for HgTe quantum wells with an inverted band structure is explained. For the band inversion to occur, the quantum wells have to exhibit a well thickness d_QW above a critical value (d_QW > d_c = 6.3 nm). Subsequently, the concept of QPCs is explicated and the corresponding transport behaviour is analytically described. Following the discussion of relevant constraints when realizing a QPC technology in a QSH system, a newly developed lithography process utilizing a multi-step wet etching technique for fabricating QPC devices based on HgTe quantum wells is presented. Transport measurements of exemplary devices show the expected conductance quantization in steps of ΔG ≈ 2e^2/h within the conduction band for a topological as well as for a trivial (d_QW < d_c) QPC. For the topological case, the residual conductance within the bulk band gap saturates at G_QSH ≈ 2e^2/h due to presence of the QSH state, while it drops to G ≈ 0 for the trivial device. Moreover, bias voltage dependent measurements of the differential conductance of an inverted sample provide explicit proof of the unperturbed coexistence of topological and trivial transport modes.
In a next step, Chapter 3 describes the emergence of a QSH interferometer state in narrow QPC devices with a quantum well thickness of d_QW = 7 nm. Presented band structure calculations reveal that the spatial extension of the QSH edge states depends on the position of the Fermi energy within the bulk band gap. As a consequence, reservoir electrons with randomized spin couple to both edge channels with the same probability under certain conditions, thus causing the formation of a QSH ring. A straightforward model capturing and specifying the occurrence of such a QSH interferometer is provided as well as substantiated by two experimental plausibility checks. After relevant quantum phases are theoretically introduced, the discussion of the obtained data reveals the accumulation of an Aharonov-Bohm phase, of a dynamical Aharonov-Casher phase as well as of a spin-orbit Berry phase of π in appropriate QPC devices. These results are consistent with analytic model considerations.
The last part of this thesis, Chapter 4, covers the observation of an unexpected conductance pattern for QPC samples fabricated from quantum wells with d_QW = 10.5 nm. In these devices, an anomalous plateau at G ≈ e^2/h = 0.5 x G_QSH emerges in addition to the QSH phase entailed residual conductance of G_QSH ≈ 2e^2/h. This so-called 0.5 anomaly occurs only for a specific interval of QPC width values, while it starts to get lost for too large sample widths. Furthermore, presented temperature and bias voltage dependent measurements insinuate that the emergence of the 0.5 anomaly is related to a gapped topological state. Additional characterization of this peculiar transport regime is provided by the realization of a novel device concept, which integrates a QPC within a standard Hall bar geometry. The results of the experimental analysis of such a sample link the occurrence of the 0.5 anomaly to a backscattered QSH channel. Thus, following a single particle perspective argumentation, it is reasoned that only one edge channel is transmitted in the context of the 0.5 anomaly. Two theoretic models possibly explaining the emergence of the 0.5 anomaly -- based on electron-electron interactions -- are discussed.
To conclude, the implementation of a working QPC technology in a QSH system represents a paramount development in the context of researching two-dimensional topological insulators and enables a multitude of future experiments. QPC devices realized in a QSH system are for example envisaged to allow for the detection of Majorana fermions and parafermions. Furthermore, the reported formation of a QSH interferometer state in appropriate QPC devices is of high interest. The observed dynamical Aharonov-Casher phase in the QSH regime enables a controllable modulation of the topological conductance, thus providing the conceptual basis for a topological transistor. Moreover, due to the resilience of geometric phases against dephasing, the presence of a spin-orbit Berry phase of π represents a promising perspective with regard to possible quantum computation concepts. Besides that, the transmission of only one QSH edge channel due to the emergence of the 0.5 anomaly is equivalent to 100 % spin polarization, which is an essential ingredient for realizing spintronic applications. Hence, the thesis at hand covers the experimental detection of three effects of fundamental importance in the context of developing new generations of logic devices -- based on QPCs fabricated from topological HgTe quantum wells.
Im Rahmen dieser Arbeit wurde ein Rastertunnelmikroskop (STM) für Messungen bei tiefen Temperaturen und hohen Magnetfeldern konzipiert und aufgebaut. Die Probentemperatur kann dabei auf bis zu 1.4\,Kelvin reduziert werden, was spektroskopische Messungen mit extrem hoher Energieauflösung ermöglicht. Die thermische Verbreiterung spektroskopischer Merkmale liegt somit im Bereich eines Milli-Elektronenvolts, wie durch den Fit der Bandlücke eines supraleitenden Materials demonstriert wird. Ein linearer Bewegungsmechanismus ermöglicht die Positionierung des STM-Körpers innerhalb einer supraleitenden Spule, in der Magnetfelder von bis zu 12.5\,Tesla senkrecht zur Probenoberfläche erzeugt werden können. Das System erlaubt des Weiteren den Wechsel von Spitzen und Proben innerhalb des Kryostaten sowie das Aufdampfen von Einzelatomen auf die kalte Probenoberfläche ohne die Probe aus dem STM zu entfernen. Um den Einfluss mechanischer Vibrationen zu minimieren wurde ein innovatives Feder-Dämpfungssystem entwickelt, dass eine Stabilität des Tunnelkontakts von bis zu einem Pikometer gewährleistet. \\
\noindent Der zweite Teil dieser Arbeit präsentiert die Ergebnisse von STM-Messungen auf Antimon-Tellurid (Sb_{2}Te_{3}). Sb_{2}Te_{3}\, gehört zur relativ neu entdeckten Materialklasse der Topologischen Isolatoren (TI). Diese Verbindungen besitzen auf ihren Oberflächen Zustände mit linearer Dispersion, die durch die Zeitumkehr-Invarianz geschützt werden. Fokus unserer Messungen ist dabei der Einfluss eines magnetischen Feldes auf die Eigenschaften eines derartigen unkonventionellen 2D-Elektronengases. Dazu wurde die Entstehung von Landau Level (LL) innerhalb eines Magnetfelds genau untersucht. Die zwei in dieser Arbeit untersuchten Hauptaspekte sind: \medskip
\noindent(i) Die energetische Verbreiterung, die Rückschlüsse auf die Lebensdauer zulässt\\
(ii) Die örtliche Fluktuation. \medskip
\noindent Erstaunlicherweise kann die gemessene Verbreiterung der Landau Resonanzen nicht mit gängigen Mechanismen der Lebenszeit-Verbreiterung erklärt werden. Aus diesem Grund wird eine alternative Interpretation basierend auf der Heissenbergschen Unschärferelation vorgestellt, die im guten Einklang mit den von uns gewonnenen Daten steht. Des Weiteren zeigen örtlich aufgelöste Messungen systematische Abweichungen in der Dirac-Geschwindigkeit positiver und negativer Landau Resonanzen. Diese Fluktuationen stehen dabei in direktem Zusammenhang mit Änderungen im lokalen chemischen Potential. Da die physikalischen Ursachen dieser Abweichung im Rahmen dieser Arbeit nicht zweifelsfrei geklärt werden konnten, werden im letzten Teil die zugrundeliegenden Messergebnisse vorgestellt und mögliche Erklärungen des Verhaltens präsentiert.
Neue Erkenntnisse über elektronische Eigenschaften von Festkörpern legen den Grundstein für innovative Anwendungen der Zukunft. Von zentraler Bedeutung sind insbesondere die Eigenschaften der Elektronenspins. Um diese besser zu verstehen, befasst sich die vorliegende Arbeit mit der experimentellen Analyse der elektronischen Struktur von topologischen Isolatoren (Sb$_2$Te$_3$ , Bi$_2$Se$_x$Te$_{3−x}$, Bi$_{1.5}$Sb$_{0.5}$Te$_{1.8}$Se$_{1.2} und Bi$_{1.4}$Sb$_{1.1}$Te$_{2.2}$S$_{0.3}$) und Kristallen mit starker Spin-Bahn-Wechselwirkung (BiTeI) mittels Photoelektronenspektroskopie. Zu Beginn werden die zum Verständnis dieser Arbeit benötigten Grundlagen erklärt sowie die unterschiedlichen zum Einsatz kommenden Techniken eingeführt. Der Hauptteil der Arbeit teilt sich in drei Forschungsschwerpunkte. Der erste Teil befasst sich mit den elektronischen Eigenschaften der Valenzbandstruktur von Sb2Te3 und den auftretenden Oberflächenzuständen. Durch gezielte Variation der Energie der anregenden Strahlung wird der Charakter der Wellenfunktion des topologischen Oberflächenzustands und dessen Wechselwirkung mit Valenzzuständen erforscht. Dabei spielt die Topologie der Volumenbandstruktur eine grundlegende Rolle. Der zusätzliche Vergleich zu Photoemissionsrechnungen ermöglicht detaillierte Einblicke in die Wechselwirkung zwischen Oberflächen- und Volumenzuständen und gibt Aufschluss darüber, wie diese vermittelt werden.
Im zweiten Abschnitt wird durch die Analyse des gemessenen Photoelektronenspins das Zusammenspiel der Spintextur des Grundzustands und Endzuständen in Bi2Te3 untersucht. Dabei treten, im Gegensatz zu Grundzustandsrechnungen, Radialkomponenten des Polarisationsvektors in nichtsymmetrischer Messgeometrie auf. Sowohl deren Energieabhängigkeit als auch deren Auftreten in Photoemissionsrechnungen (1-Schritt-Modell) deutet darauf hin, dass diese ihren Ursprung in Übergangsmatrixelementen des Photoemissionsprozesses haben. Dieses Ergebnis wird mit Spinpolarisationsmessungen am Oberflächenzustand des nicht-topologischen Schichtsystems BiTeI verglichen.
Im dritten Teil werden Auswirkungen unterschiedlicher Manipulationen der untersuchten Materialien auf deren elektronische Eigenschaften beschrieben. Die Adsorption von Bruchteilen einer monoatomaren Lage des Alkalimetalls Caesium auf die Oberfläche des topologischen Isolators Sb2Te3 wird systematisch untersucht. Dadurch kann dessen intrinsische p-Dotierung teilweise abgebaut werden, wobei die Valenzbandstruktur trotz der Reaktivität des Adsorbats intakt bleibt. Des Weiteren werden Auswirkungen von Änderungen der Kristallstöchiometrie durch Volumendotierung vergleichend diskutiert.
Ausblickend befasst sich das Kapitel mit dem Verhalten geringer Mengen ferromagnetischer
Materialen (Fe, Ni) auf den Oberflächen der topologischen Isolatoren. Für die verschiedenen Adsorbate werden Trends aufgezeigt, die von Temperatur und Zusammensetzung des Substratkristalls abhängen.
Molecular Beam Epitaxy and Characterization of Bi-Based V\(_2\)VI\(_3\) Topological Insulators
(2016)
The present thesis is addressed to the growth and characterization of Bi-based V2VI3 topological insulators (TIs). The TIs were grown by molecular beam epitaxy (MBE) on differently passivated Si(111) substrates, as well as InP(111) substrates. This allows the study of the influence of the substrate on the structural and electrical properties of the TIs.
The Bi2Se3 layers show a change of mosaicity-tilt and -twist for growth on the differently prepared Si(111) substrates, as well as a significant increase of crystalline quality for growth on the lateral nearly lattice matched InP(111). The rocking curve FWHMs observed for thick layers grown on InP are comparable to these of common zincblende layers, which are close to the resolution limit of standard high resolution X-ray diffraction (HRXRD) setups. The unexpected high structural crystalline quality achieved in this material system is remarkable due to the presence of weak van der Waals bonds between every block of five atomic layers, i.e. a quintuple layer (QL), in growth direction.
In addition to the mosaicity also twin domains, present in films of the V2VI3 material system, are studied. The twin defects are observed in Bi2Se3 layers grown on Si(111) and lattice matched InP(111) suggesting that the two dimensional surface lattice of the substrates can not determine the stacking order ABCABC... or ACBACB... in locally separated growth seeds. Therefore the growth on misoriented and rough InP(111) is analyzed.
The rough InP(111) with its facets within a hollow exceeding the height of a QL is able to provide its stacking information to the five atomic layers within a QL. By varying the roughness of the InP substrate surface, due to thermal annealing, the influence on the twinning within the layer is confirmed resulting in a complete suppression of twin domains on rough InP(111).
Focusing on the electrical properties of the Bi2Se3 films, the increased structural quality for films grown on lattice matched flat InP(111)B results in a marginal reduction of carrier density by about 10% compared to the layers grown on H-passivated Si(111), whereas the suppression of twin domains for growth on rough InP(111)B resulted in a reduction of carrier density by an order of magnitude. This implies, that the twin domains are a main crystal defect responsible for the high carrier density in the presented Bi2Se3 thin films.
Besides the binary Bi2Se3 also alloys with Sb and Te are fabricated to examine the influence of the compound specific point defects on the carrier density. Therefore growth series of the ternary materials Bi2Te(3-y)Se(y), Bi(2-x)Sb(x)Se3, and Bi(2-x)Sb(x)Te3, as well as the quaternary Bi(2-x)Sb(x)Te(3-y)Se(y) are studied.
To further reduce the carrier density of twin free Bi2Se3 layers grown on InP(111)B:Fe a series of Bi(2-x)Sb(x)Se3 alloys were grown under comparable growth conditions. This results in a reduction of the carrier density with a minimum in the composition range of about x=0.9-1.0.
The Bi(2-x)Sb(x)Te3 alloys exhibit a pn-transition, due to the dominating n-type and p-type point defects in its binary compounds, which is determined to reduce the bulk carrier density enabling the study the TI surface states. This pn-transition plays a significant role in realizing predicted applications and exotic effects, such as the quantum anomalous Hall effect.
The magnetic doping of topological insulators with transition metals is studied by incorporating Cr and V in the alloy Bi(2-x)Sb(x)Te3 by codeposition. The preferential incorporation of Cr on group-V sites is confirmed by EDX and XRD, whereas the incorporation of Cr reduces the crystalline quality of the layer. Magnetotransport measurements of the Cr-doped TIs display an anomalous Hall effect confirming the realization of a magnetic TI thin film. The quantum anomalous Hall effect is observed in V-doped Bi(2-x)Sb(x)Te3, where the V-doping results in higher Curie temperatures, as well as higher coercive fields compared to the Cr-doping of the TIs.
Moreover the present thesis contributes to the understanding of the role of the substrate concerning the crystalline quality of van der Waals bonded layers, such as the V2VI3 TIs, MoS2 and WoTe2. Furthermore, the fabrication of the thin film TIs Bi(2-x)Sb(x)Te(3-y)Se(y) in high crystalline quality serves as basis to explore the physics of topological insulators.
In the past few years, two-dimensional quantum liquids with fractional excitations have been a topic of high interest due to their possible application in the emerging field of quantum computation and cryptography. This thesis is devoted to a deeper understanding of known and new fractional quantum Hall states and their stabilization in local models. We pursue two different paths, namely chiral spin liquids and fractionally quantized, topological phases.
The chiral spin liquid is one of the few examples of spin liquids with fractional statistics. Despite its numerous promising properties, the microscopic models for this state proposed so far are all based on non-local interactions, making the experimental realization challenging. In the first part of this thesis, we present the first local parent Hamiltonians, for which the Abelian and non-Abelian chiral spin liquids are the exact and, modulo a topological degeneracy, unique ground states. We have developed a systematic approach to find an annihilation operator of the chiral spin liquid and construct from it a many-body interaction which establishes locality. For various system sizes and lattice geometries, we numerically find largely gapped eigenspectra and confirm to an accuracy of machine precision the uniqueness of the chiral spin liquid as ground state of the respective system. Our results provide an exact spin model in which fractional quantization can be studied.
Topological insulators are one of the most actively studied topics in current condensed matter physics research. With the discovery of the topological insulator, one question emerged: Is there an interaction-driven set of fractionalized phases with time reversal symmetry? One intuitive approach to the theoretical construction of such a fractional topological insulator is to take the direct product of a fractional quantum Hall state and its time reversal conjugate. However, such states are well studied conceptually and do not lead to new physics, as the idea of taking a state and its mirror image together without any entanglement between the states has been well understood in the context of topological insulators. Therefore, the community has been looking for ways to implement some topological interlocking between different spin species. Yet, for all practical purposes so far, time reversal symmetry has appeared to limit the set of possible fractional states to those with no interlocking between the two spin species.
In the second part of this thesis, we propose a new universality class of fractionally quantized, topologically ordered insulators, which we name “fractional insulator”. Inspired by the fractional quantum Hall effect, spin liquids, and fractional Chern insulators, we develop a wave function approach to a new class of topological order in a two-dimensional crystal of spin-orbit coupled electrons. The idea is simply to allow the topological order to violate time reversal symmetry, while all locally observable quantities remain time reversal invariant. We refer to this situation as “topological time reversal symmetry breaking”. Our state is based on the Halperin double layer states and can be viewed as a two-layer system of an ↑-spin and a ↓-spin sphere. The construction starts off with Laughlin states for the ↑-spin and ↓-spin electrons and an interflavor term, which creates correlations between the two layers. With a careful parameter choice, we obtain a state preserving time reversal symmetry locally, and label it the “311-state”. For systems of up to six ↑-spin and six ↓-spin electrons, we manage to construct an approximate parent Hamiltonian with a physically realistic, local interaction.
The subject of this thesis is the investigation of the transport properties of topological and massive surface states in the three-dimensional topological insulator Hg(Mn)Te. These surface states give rise to a variety of extraordinary transport phenomena, making this material system of great interest for research and technological applications. In this connection, many physical properties of the topological insulator Hg(Mn)Te still require in-depth exploration. The overall aim of this thesis is to analyze the quantum transport of HgTe-based devices ranging from hundreds of micrometers (macroscopic) down to a few micrometers in size (microscopic) in order to extend the overall understanding of surface states and the possibilities of their manipulation.
In order to exploit the full potential of our high-quality heterostructures, it was necessary to revise and improve the existing lithographic fabrication process of macroscopic three-dimensional Hg(Mn)Te samples. A novel lithographic standard recipe for the fabrication of the HgTe-based macrostructures was developed. This recipe includes the use of an optimized Hall bar design and wet etching instead of etching with high-energy \(\mathrm{{Ar^{+}}}\)-ions, which can damage the samples. Further, a hafnium oxide insulator is applied replacing the SiO\(_{2}\)/Si\(_{3}\)N\(_{4}\) dielectric in order to reduce thermal load. Moreover, the devices are metallized under an alternating angle to avoid discontinuities of the metal layers over the mesa edges. It was revealed that the application of gate-dielectric and top-gate metals results in n-type doping of the devices. This phenomenon could be attributed to quasi-free electrons tunneling from the trap states, which form at the interface cap layer/insulator, through the cap into the active layer. This finding led to the development of a new procedure to characterize wafer materials. It was found that the optimized lithographic processing steps do not unintentionally react chemically with our heterostructures, thus avoiding a degradation of the quality of the Hg(Mn)Te layer. The implementation of new contact structures Ti/Au, In/Ti/Au, and Al/Ti/Au did not result in any improvement compared to the standard structure AuGe/Au. However, a novel sample recipe could be developed, resulting in an intermixing of the contact metals (AuGe and Au) and fingering of metal into the mesa. The extent of the quality of the ohmic contacts obtained through this process has yet to be fully established.
This thesis further deals with the lithographic realization of three-dimensional HgTe-based microstructures measuring only a few micrometer in size. Thus, these structures are in the order of the mean free path and the spin relaxation length of topological surface state electrons. A lithographic process was developed enabling the fabrication of nearly any desired microscopic device structure. In this context, two techniques suitable for etching microscopic samples were realized, namely wet etching and the newly established inductively coupled plasma etching. While wet etching was found to preserve the crystal quality of the active layer best, inductively coupled plasma etching is characterized by high reproducibility and excellent structural fidelity. Hence, the etching technique employed depends on the envisaged type of experiment.
Magneto-transport measurements were carried out on the macroscopic HgTe-based devices fabricated by means of improved lithographic processing with respect to the transport properties of topological and massive surface states. It was revealed that due to the low charge carrier density present in the leads to the ohmic contacts, these regions can exhibit an insulating behavior at high magnetic fields and extremely low temperatures. As soon as the filling factor of the lowest Landau levels dropped below a critical value (\(\nu_{\mathrm{{c}}}\approx0.8\)), the conductance of the leads decreased significantly. It was demonstrated that the carrier density in the leads can be increased by the growth of modulation doping layers, a back-gate-electrode, light-emitting diode illumination, and by the application of an overlapping top-gate layout. This overlapping top-gate and a back-gate made it possible to manipulate the carrier density of the surface states on both sides of the Hg(Mn)Te layer independently. With this setup, it was identified that topological and massive surface states contribute to transport simultaneously in 3D Hg(Mn)Te. A model could be developed allowing the charge carrier systems populated in the sample to be determined unambiguously. Based on this model, the process of the re-entrant quantum Hall effect observed for the first time in three-dimensional topological insulators could be explained by an interplay of n-type topological and p-type massive surface states. A well-pronounced \(\nu=-1\rightarrow\nu=-2\rightarrow\nu=-1\) sequence of quantum Hall plateaus was found in manganese-doped HgTe-based samples. It is postulated that this is the condensed-matter realization of the parity anomaly in three-dimensional topological insulators. The actual nature of this phenomenon can be the subject of further research. In addition, the measurements have shown that inter-scattering occurs between counter-propagating quantum Hall edge states. The good quantization of the Hall conductance despite this inter-scattering indicates that only the unpaired edge states determine the transport properties of the system as a whole. The underlying inter-scattering mechanism is the topic of a publication in preparation.
Furthermore, three-dimensional HgTe-based microstructures shaped like the capital letter "H" were investigated regarding spin transport phenomena. The non-local voltage signals occurring in the measurements could be attributed to a current-induced spin polarization of the topological surface states due to electrons obeying spin-momentum locking. It was shown that the strength of this non-local signal is directly connected to the magnitude of the spin polarization and can be manipulated by the applied top-gate voltage. It was found that in these microstructures, the massive surface and bulk states, unlike the topological surface states, cannot contribute to this spin-associated phenomenon. On the contrary, it was demonstrated that the population of massive states results in a reduction of the spin polarization, either due to the possible inter-scattering of massive and topological surface states or due to the addition of an unpolarized electron background. The evidence of spin transport controllable by a top-gate-electrode makes the three-dimensional material system mercury telluride a promising candidate for further research in the field of spintronics.
In the field of spintronics, spin manipulation and spin transport are the main principles that need to be implemented. The main focus of this thesis is to analyse semiconductor systems where high fidelity in these principles can be achieved. To this end, we use numerical methods for precise results, supplemented by simpler analytical models for interpretation.
The material system of 2D topological insulators, HgTe/CdTe quantum wells, is interesting not only because it provides a topologically distinct phase of matter, physically manifested in its protected transport properties, but also since within this system, ballistic transport of high quality can be realized, with Rashba spin-orbit coupling and electron densities that are tunable by electrical gating. Extending the Bernvevig-Hughes-Zhang model for 2D topological insulators, we derive an effective four-band model including Rashba spin-orbit terms due to an applied potential that breaks the spatial inversion symmetry of the quantum well. Spin transport in this system shows interesting physics because the effects of Rashba spin-orbit terms and the intrinsic Dirac-like spin-orbit terms compete. We show that the resulting spin Hall signal can be dominated by the effect of Rashba spin-orbit coupling. Based on spin splitting due to the latter, we propose a beam splitter setup for all-electrical generation and detection of spin currents. Its working principle is similar to optical birefringence. In this setup, we analyse spin current and spin polarization signals of different spin vector components and show that large in-plane spin polarization of the current can be obtained. Since spin is not a conserved quantity of the model,
we first analyse the transport of helicity, a conserved quantity even in presence of Rashba spin-orbit terms. The polarization defined in terms of helicity is related to in-plane polarization of the physical spin.
Further, we analyse thermoelectric transport in a setup showing the spin Hall effect. Due to spin-orbit coupling, an applied temperature gradient generates a transverse spin current, i.e. a spin Nernst effect, which is related to the spin Hall effect by a Mott-like relation. In the metallic energy regimes, the signals are qualitatively explained by simple analytic models. In the insulating regime, we observe a spin Nernst signal that originates from the finite-size induced overlap of edge states.
In the part on methods, we discuss two complementary methods for construction of effective semiconductor models, the envelope function theory and the method of invariants. Further, we present elements of transport theory, with some emphasis on spin-dependent signals. We show the connections of the adiabatic theorem of quantum mechanics to the semiclassical theory of electronic transport and to the characterization of topological phases. Further, as application of the adiabatic theorem to a control problem, we show that universal control of a single spin in a heavy-hole quantum dot is experimentally realizable without breaking time reversal invariance,
but using a quadrupole field which is adiabatically changed as control knob. For experimental realization, we propose a GaAs/GaAlAs quantum well system.
Emergent phenomena in condensed matter physics like, e.g., magnetism, superconductivity, or non-trivial topology often come along with a surprise and exert great fascination to researchers up to this day. Within this thesis, we are concerned with the analysis of associated types of order that arise due to strong electronic interactions and focus on the high-\(T_c\) cuprates and Kondo systems as two prime candidates. The underlying many-body problem cannot be solved analytically and has given rise to the development of various approximation techniques to tackle the problem.
In concrete terms, we apply the auxiliary particle approach to investigate tight-binding Hamiltonians subject to a Hubbard interaction term to account for the screened Coulomb repulsion. Thereby, we adopt the so-called Kotliar-Ruckenstein slave-boson representation that reduces the problem to non-interacting quasiparticles within a mean-field approximation. Part I provides a pedagogical review of the theory and generalizes the established formalism to encompass Gaussian fluctuations around magnetic ground states as a crucial step to obtaining novel results.
Part II addresses the two-dimensional one-band Hubbard model, which is known to approximately describe the physics of the high-\(T_c\) cuprates that feature high-temperature superconductivity and various other exotic quantum phases that are not yet fully understood. First, we provide a comprehensive slave-boson analysis of the model, including the discussion of incommensurate magnetic phases, collective modes, and a comparison to other theoretical methods that shows that our results can be massively improved through the newly implemented fluctuation corrections. Afterward, we focus on the underdoped regime and find an intertwining of spin and charge order signaled by divergences of the static charge susceptibility within the antiferromagnetic domain. There is experimental evidence for such inhomogeneous phases in various cuprate materials, which has recently aroused interest because such correlations are believed to impact the formation of Cooper pairs. Our analysis identifies two distinct charge-ordering vectors, one of which can be attributed to a Fermi-surface nesting effect and quantitatively fits experimental data in \(\mathrm{Nd}_{2-\mathrm{x}}\mathrm{Ce}_\mathrm{x}\mathrm{CuO}_4\) (NCCO), an electron-doped cuprate compound. The other resembles the so-called Yamada relation implying the formation of periodic, double-occupied domain walls with a crossover to phase separation for small dopings.
Part III investigates Kondo systems by analyzing the periodic Anderson model and its generalizations. First, we consider Kondo metals and detect weakly magnetized ferromagnetic order in qualitative agreement with experimental observations, which hinders the formation of heavy fermions. Nevertheless, we suggest two different parameter regimes that could host a possible Kondo regime in the context of one or two conduction bands. The part is concluded with the study of topological order in Kondo insulators based on a three-dimensional model with centrosymmetric spin-orbit coupling. Thereby, we classify topologically distinct phases through appropriate \(\mathbb{Z}_2\) invariants and consider paramagnetic and antiferromagnetic mean-field ground states. Our model parameters are chosen to specifically describe samarium hexaboride (\(\mbox{SmB}_6\)), which is widely believed to be a topological Kondo insulator, and we identify topologically protected surface states in agreement with experimental evidence in that material. Moreover, our theory predicts the emergence of an antiferromagnetic topological insulator featuring one-dimensional hinge-states as the signature of higher-order topology in the strong coupling regime. While the nature of the true ground state is still under debate, corresponding long-range magnetic order has been observed in pressurized or alloyed \(\mbox{SmB}_6\), and recent experimental findings point towards non-trivial topology under these circumstances. The ability to understand and control topological systems brings forth promising applications in the context of spintronics and quantum computing.
Realization and Spectroscopy of the Quantum Spin Hall Insulator Bismuthene on Silicon Carbide
(2022)
Topological matter is one of the most vibrant research fields of contemporary solid state physics since the theoretical prediction of the quantum spin Hall effect in graphene in 2005. Quantum spin Hall insulators possess a vanishing bulk conductivity but symmetry-protected, helical edge states that give rise to dissipationless charge transport.
The experimental verification of this exotic state of matter in 2007 lead to a boost of research activity in this field, inspired by possible ground-breaking future applications.
However, the use of the quantum spin Hall materials available to date is limited to cryogenic temperatures owing to their comparably small bulk band gaps.
In this thesis, we follow a novel approach to realize a quantum spin Hall material with a large energy gap and epitaxially grow bismuthene, i.e., Bi atoms adopting a honeycomb lattice, in a \((\sqrt{3}\times\sqrt{3})\) reconstruction on the semiconductor SiC(0001). In this way, we profit both from the honeycomb symmetry as well as the large spin-orbit coupling of Bi, which, in combination, give rise to a topologically non-trivial band gap on the order of one electronvolt.
An in-depth theoretical analysis demonstrates that the covalent bond between the Si and Bi atoms is not only stabilizing the Bi film but is pivotal to attain the quantum spin Hall phase.
The preparation of high-quality, unreconstructed SiC(0001) substrates sets the basis for the formation of bismuthene and requires an extensive procedure in ultra-pure dry H\(_2\) gas. Scanning tunneling microscopy measurements unveil the (\(1\times1\)) surface periodicity and smooth terrace planes, which are suitable for the growth of single Bi layers by means of molecular beam epitaxy. The chemical configuration of the resulting Bi film and its oxidation upon exposure to ambient atmosphere are inspected with X-ray photoelectron spectroscopy.
Angle-resolved photoelectron spectroscopy reveals the excellent agreement of probed and calculated band structure. In particular, it evidences a characteristic Rashba-splitting of the valence bands at the K point. Scanning tunneling spectroscopy probes signatures of this splitting, as well, and allows to determine the full band gap with a magnitude of \(E_\text{gap}\approx0.8\,\text{eV}\).
Constant-current images and local-density-of-state maps confirm the presence of a planar honeycomb lattice, which forms several domains due to different, yet equivalent, nucleation sites of the (\(\sqrt{3}\times\sqrt{3}\))-Bi reconstruction.
Differential conductivity measurements demonstrate that bismuthene edge states evolve at atomic steps of the SiC substrate. The probed, metallic local density of states is in agreement with the density of states expected from the edge state's energy dispersion found in density functional theory calculations - besides a pronounced dip at the Fermi level.
By means of temperature- and energy-dependent tunneling spectroscopy it is shown that the spectral properties of this suppressed density of states are successfully captured in the framework of the Tomonaga-Luttinger liquid theory and most likely originate from enhanced electronic correlations in the edge channel.
Over the last decade, the field of topological insulators has become one of the most vivid areas in solid state physics. This novel class of materials is characterized by an insulating bulk gap, which, in two-dimensional, time-reversal symmetric systems, is closed by helical edge states. The latter make topological insulators promising candidates for applications in high fidelity spintronics and topological quantum computing. This thesis contributes to bringing these fascinating concepts to life by analyzing transport through heterostructures formed by two-dimensional topological insulators in contact with metals or superconductors. To this end, analytical and numerical calculations are employed. Especially, a generalized wave matching approach is used to describe the edge and bulk states in finite size tunneling junctions on the same footing.
The numerical study of non-superconducting systems focuses on two-terminal metal/topological
insulator/metal junctions. Unexpectedly, the conductance signals originating from the bulk and
the edge contributions are not additive. While for a long junction, the transport is determined
purely by edge states, for a short junction, the conductance signal is built from both bulk and
edge states in a ratio, which depends on the width of the sample. Further, short junctions show
a non-monotonic conductance as a function of the sample length, which distinguishes the topologically non-trivial regime from the trivial one. Surprisingly, the non-monotonic conductance of the topological insulator can be traced to the formation of an effectively propagating solution, which is robust against scalar disorder.
The analysis of the competition of edge and bulk contributions in nanostructures is extended to transport through topological insulator/superconductor/topological insulator tunneling junctions. If the dimensions of the superconductor are small enough, its evanescent bulk modes
can couple edge states at opposite sample borders, generating significant and tunable crossed
Andreev reflection. In experiments, the latter process is normally disguised by simultaneous
electron transmission. However, the helical edge states enforce a spatial separation of both competing processes for each Kramers’ partner, allowing to propose an all-electrical measurement
of crossed Andreev reflection.
Further, an analytical study of the hybrid system of helical edge states and conventional superconductors in finite magnetic fields leads to the novel superconducting quantum spin Hall effect. It is characterized by edge states. Both the helicity and the protection against scalar disorder of these edge states are unaffected by an in-plane magnetic field. At the same time its superconducting gap and its magnetotransport signals can be tuned in weak magnetic fields, because the combination of helical edge states and superconductivity results in a giant g-factor. This is manifested in a non-monotonic excess current and peak splitting of the dI/dV characteristics as a function of the magnetic field. In consequence, the superconducting quantum spin Hall effect is an effective generator and detector for spin currents.
The research presented here deepens the understanding of the competition of bulk and edge
transport in heterostructures based on topological insulators. Moreover it proposes feasible experiments to all-electrically measure crossed Andreev reflection and to test the spin polarization of helical edge states.
Topological insulators are electronic phases that insulate in the bulk and accommodate a peculiar, metallic edge liquid with a spin-dependent dispersion.
They are regarded to be of considerable future use in spintronics and for quantum computation.
Besides determining the intrinsic properties of this rather novel electronic phase, considering its combination with well-known physical systems can generate genuinely new physics.
In this thesis, we report on such combinations including topological insulators. Specifically, we analyze an attached Rashba impurity, a Kondo dot in the two channel setup, magnetic impurities on the surface of a strong three-dimensional topological insulator, the proximity coupling of the latter system to a superconductor, and hybrid systems consisting of a topological insulator and a semimetal.
Let us summarize our primary results.
Firstly, we determine an analytical formula for the Kondo cloud and describe its possible detection in current correlations far away from the Kondo region.
We thereby rely on and extend the method of refermionizable points.
Furthermore, we find a class of gapless topological superconductors and semimetals, which accommodate edge states that behave similarly to the ones of globally gapped topological phases. Unexpectedly, we also find edge states that change their chirality when affected by sufficiently strong disorder.
We regard the presented research helpful in future classifications and applications of systems containing topological insulators, of which we propose some examples.
The thesis at hand is concerned with improving our understanding of and our control over transport properties of the three-dimensional topological insulator HgTe. Topological insulators are characterized by an insulating bulk and symmetry-protected metallic surface states. These topological surface states hold great promise for research and technology; at the same time, many properties of experimentally accessible topological insulator materials still need to be explored thoroughly. The overall aim of this thesis was to experimentally investigate micrometer-sized HgTe transport devices to observe the ballistic transport regime as well as intercarrier scattering and possibly identify special properties of the topological surface states.
Part I of the thesis presents lithographic developments concerned with etching small HgTe devices. The aim was to replace existing processes which relied on dry etching with high-energy \(\text{Ar}^+\) ions and an organic etch mask. This etching method is known to degrade the HgTe crystal quality. In addition, the etch mask turned out to be not durable for long etching processes and difficult to remove completely after etching. First, \(\text{BaF}_2\) was introduced as a new etch mask for dry etching to replace the organic etch mask. With common surface characterization techniques like SEM and XPS it was shown that \(\text{BaF}_2\) etch masks are easy to deposit, highly durable in common dry etching processes for \(\text{Hg}_{1-x}\text{Cd}_x\text{Te}\), and easy to remove in deionized water. Transport results of HgTe devices fabricated with the new etch mask are comparable to results obtained with the old process. At the same time, the new etch mask can withstand longer etching times and does not cause problems due to incomplete removal. Second, a new inductively coupled plasma dry etching process based on \(\text{CH}_4\) and Ar was introduced. This etching process is compatible with \(\text{BaF}_2\) etch masks and yields highly reproducible results. Transport results indicate that the new etching process does not degrade the crystal quality and is suitable to produce high-quality transport devices even in the micrometer range. A comparison with wet-etched samples shows that inductively coupled plasma etching introduces a pronounced edge roughness. This - usually undesirable - property is actually beneficial for some of the experiments in this study and mostly irrelevant for others. Therefore, most samples appearing in this thesis were fabricated with the new process.
Part II of the thesis details the advancements made in identifying topological and trivial states which contribute to transport in HgTe three-dimensional topological insulators. To this end, macroscopic Hall bar samples were fabricated from high-quality tensilely strained HgTe layers by means of the improved lithographic processes. All samples were equipped with a top gate electrode, and some also with a modulation doping layer or a back gate electrode to modify the carrier density of the surface states on both sides of the HgTe layer. Due to the high sample quality, Landau levels could be well-resolved in standard transport measurements down to magnetic fields of less than 0.5T. High-resolution measurements of the Landau level dispersion with gate voltage and magnetic field allowed disentangling different transport channels. The main result here is that the upper (electron) branches of the two topological surface states contribute to transport in all experimentally relevant density regimes, while the hole branch is not accessible. Far in n-regime bulk conduction band states give a minor contribution to transport. More importantly, trivial bulk valence band holes come into play close to the charge neutrality point. Further in p-regime, the strong applied gate voltage leads to the formation of two-dimensional, massive hole states at the HgTe surface. The interplay of different states gives rise to rich physics: Top gate-back gate maps revealed that an anticrossing of Landau levels from the two topological surface states occurs at equal filling. A possible explanation for this effect is a weak hybridization of the surface states; however, future studies need to further clarify this point. Furthermore, the superposition of n-type topological and p-type trivial surface states leads to an intriguing Landau level dispersion. The good quantization of the Hall conductance in this situation indicates that the counterpropagating edge states interact with each other. The nature of this interaction will be the topic of further research.
Part III of the thesis is focused on HgTe microstructures. These "channel samples" have a typical width of 0.5 to 4µm and a typical length of 5 to 80µm. The quality of these devices benefits particularly from the improved lithographic processes. As a result, the impurity mean free path of the topological surface state electrons is on the order of the device width and transport becomes semiballistic. This was verified by measuring the channel resistance in small magnetic fields in n-regime. The deflection of carriers towards the dissipative channel walls results in a pronounced peak in the magnetoresistance, which scales in a predictable manner with the channel width. To investigate transport effects due to mutual scattering of charge carriers, the differential resistance of channel samples was measured as a function of carrier temperature. Selective heating of the charge carriers - but not the lattice - was achieved by passing a heating current through the channel. Increasing the carrier temperature has two pronounced effects when the Fermi level is situated in proximity to the bulk valence band maximum where the density of states is large. First, when both topological surface state electrons and bulk holes are present, electron-hole scattering leads to a pronounced increase in resistance with increasing carrier temperature. Second, a thermally induced increase of the electron and hole carrier densities reduces the resistance again at higher temperatures. A model considering these two effects was developed, which can well reproduce the experimental results. Current heating experiments in zero-gap HgTe quantum wells and compressively strained HgTe layers are consistent with this model. These observations raise the question as to how electron-hole scattering may affect other transport properties of HgTe-based three-dimensional topological insulators, which is briefly discussed in the outlook.
Topological insulators belong to a new quantum state of matter that is currently one of
the most recognized research fields in condensed matter physics. Strained bulk HgTe
and HgTe/HgCdTe quantum well structures are currently one of few topological insulator
material systems suitable to be studied in transport experiments. In addition
HgTe quantum wells provide excellent requirements for the conduction of spintronic
experiments. A fundamental requirement for most experiments, however, is to reliably
pattern these heterostructures into advanced nano-devices. Nano-lithography on this
material system proves to be challenging because of inherent temperature limitations,
its high reactivity with various metals and due to its properties as a topological insulator.
The current work gives an insight into why many established semiconductor
lithography processes cannot be easily transferred to HgTe while providing alternative
solutions. The presented developments include novel ohmic contacts, the prevention
of metal sidewalls and redeposition fences in combination with low temperature
(80 °C) lithography and an adapted hardmask lithography process utilizing a sacrificial
layer. In addition we demonstrate high resolution low energy (2.5 kV) electron beam
lithography and present an alternative airbridge gating technique. The feasibility of
nano-structures on HgTe quantum wells is exemplarily verified in two separate transport
experiments. We are first to realize physically etched quantum point contacts
in HgTe/HgCdTe high mobility 2DEGs and to prove their controllability via external
top-gate electrodes. So far quantum point contacts have not been reported in TI
materials. However, these constrictions are part of many proposals to probe the nature
of the helical quantum spin Hall edge channels and are suggested as injector and
detector devices for spin polarized currents. To confirm their functionality we performed
four-terminal measurements of the point contact conductance as a function of
external gate voltage. Our measurements clearly exhibit quantized conductance steps
in 2e2/h, which is a fundamental characteristic of quantum point contacts. Furthermore
we conducted measurements on the formation and control of collimated electron beams, a key feature to realize an all electrical spin-optic device. In a second study
several of the newly developed lithography techniques were implemented to produce
arrays of nano-wires on inverted and non-inverted HgTe quantum well samples. These
devices were used in order to probe and compare the weak antilocalization (WAL) in
these structures as a function of magnetic field and temperature. Our measurements
reveal that the WAL is almost an order of magnitude larger in inverted samples. This
observation is attributed to the Dirac-like dispersion of the energy bands in HgTe quantum
wells. The described lithography has already been successfully implemented and
adapted in several published studies. All processes have been optimized to guarantee
a minimum effect on the heterostructure’s properties and the sample surface, which is
especially important for probing the topological surface states of strained HgTe bulk
layers. Our developments therefore serve as a base for continuous progress to further
establish HgTe as a topological insulator and give access to new experiments.