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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.
Recently a new state of matter was discovered in which the bulk insulating state in a material is accompanied by conducting surface or edge states. This new state of matter can be distinguished from a conventional insulator phase by the topological properties of its band structure which led to the name "topological insulators". Experimentally, topological insulator states are mostly found in systems characterized by a band inversion compared to conventional systems. In most topological insulator systems, this is caused by a combination of energetically close bands and spin orbit coupling. Such properties are found in systems with heavy elements like Hg and Bi. And indeed, the first experimental discovery of a topological insulator succeeded in HgTe quantum wells and later also in BiSb bulk systems.
Topological insulators are of large interest due to their unique properties: In 2-dimensional topological insulators one dimensional edge states form without the need of an external magnetic field (in contrast to the quantum Hall effect). These edge states feature a linear band dispersion, a so called Dirac dispersion. The quantum spin Hall states are helical edge states, which means they consist of counterpropagating oppositely spin polarized edge channels. They are therefore of great potential for spintronic applications as well as building blocks for new more exotic states like Majorana Fermions. 3-dimensional topological insulators feature 2-dimensional surface states with only one Dirac band (also called Dirac cone) on each surface and an interesting spin texture where spin and momentum are locked perpendicular to each other in the surface plane. This unique surface band structure is predicted to be able to host several exotic states like e.g. Majorana Fermions (in combination with superconductors) and magnetic monopole like excitations.
This PhD thesis will summarize the discovery of topological insulators and highlights the developments on their experimental observations. The work focuses on HgTe which is up to now the only topological insulator material where the expected properties are unambiguously demonstrated in transport experiments. In HgTe, the topological insulator properties arise from the inversion of the Gamma_6 and Gamma_8 bands. The band inversion in HgTe is due to a combination of a high spin orbit splitting in Te and large energy corrections (due to the mass-velocity term) to the energy levels in Hg. Bulk HgTe, however, is a semimetal, which means for the conversion into a topological insulator a band gap has to be opened. In two dimensions (HgTe quantum well structures) this is achieved via quantum confinement, which opens a band gap between the quantum well subbands. In three dimensions, strain is used to lift the degeneracy of the semimetallic Gamma_8 bands opening up a band gap.
The thesis is structured as follows:
- The first chapter of this thesis will give a brief overview on discoveries in the field of topological insulators. It focuses on works relevant to experimental results presented in the following chapters. This includes a short outline of the early predictions and a summary of important results concerning 2-dimensional topological insulators while the final section discusses observations concerning 3-dimensional topological insulators.
- The discovery of the quantum spin Hall effect in HgTe marked the first experimental observation of a topological insulator. Chapter 2 will focus on HgTe quantum wells and the quantum spin Hall effect.
Above a critical thickness, HgTe quantum wells are predicted to host the quantum spin Hall state, the signature of a 2-dimensional topological insulator. HgTe quantum wells exhibiting low carrier concentrations and at the same time high carrier mobilities are required to be able to measure the quantum spin Hall effect. The growth of such high quality HgTe quantum wells was one of the major goals for this work. Continuous optimization of the substrate preparation and growth conditions resulted in controlled carrier densities down to a few 10^10 cm^-2. At the same time, carrier mobilities exceeding 1 x 10^6 cm^2/Vs have been achieved, which provides mean free paths of several micrometers in the material. Thus the first experimental evidence for the existence of the quantum spin Hall edge states succeeded in transport experiments on microstructures: When the Fermi energy was located in the bulk band gap a residual quantized resistance of 2e^2/h was found. Further experiments focused on investigating the nature of transport in this regime. By non-local measurements the edge state character could be established. The measured non-local resistances corresponded well with predictions from the Landauer-Büttiker theory applied to transport in helical edge channels.
In a final set of experiments the spin polarization of the edge channels was investigated. Here, we could make use of the advantage that HgTe quantum well structures exhibit a large Rashba spin orbit splitting. In systems with a large Rashba spin orbit splitting a spin accumulation is expected to occur at the edge of the sample perpendicular to a current flow. This so-called spin Hall effect was then used as a spin injector and detector. Using split gate devices it was possible to bring spin Hall and quantum spin Hall state into direct contact, which enabled an all electrical detection of the spin polarization of the quantum spin Hall edge channels.
- HgTe as a 3-dimensional topological insulator will be presented in chapter 3. Straining the HgTe layer enables the observation of topological insulator behavior. It was found that strain can be easily implemented during growth by using CdTe substrates. CdTe has a slightly larger lattice constant than HgTe and therefore leads to tensile strain in the HgTe layer as long as the growth is pseudomorphic. Magnetotransport studies showed the emergence of quantum Hall transport with characteristic signatures of a Dirac type bandstructure. Thus, this result marks the first observation of the quantum Hall effect in the surface states of a 3-dimensional topological insulator.
Transport experiments on samples fitted with a top gate enabled the identification of contributions from individual surfaces. Furthermore, the surface state quantum Hall effect was found to be surprisingly stable, perturbations due to additional bulk transport could not be found, even at high carrier densities of the system.
- Chapters 4 - 6 serve as in depth overviews of selected works: Chapter 4 presents a detailed overview on the all electrical detection of the spin Hall effect in HgTe quantum wells. The detection of the spin polarization of the quantum spin Hall effect is shown in chapter 5 and chapter 6 gives a detailed overview on the quantum Hall effect originating from the topological surface state in strained bulk HgTe.
The investigations discussed in this thesis pioneered the experimental work on the transport properties of topological insulator systems. The understanding of the fundamental properties of topological insulators enables new experiments in which e.g. the inclusion of magnetic dopants or the interplay between topological insulator and superconductors can be investigated in detail.
A novel growth method has been developed, allowing for the growth of strained HgTe shells on CdTe nanowires (NWs). The growth of CdTe-HgTe core-shell NWs required high attention in controlling basic parameters like substrate temperature and the intensity of supplied material fluxes. The difficulties in finding optimized growth conditions have been successfully overcome in this work.
We found the lateral redistribution of liquid growth seeds with a ZnTe growth start to be crucial to trigger vertical CdTe NW growth. Single crystalline zinc blende CdTe NWs grew, oriented along [111]B. The substrate temperature was the most critical parameter to achieve straight and long wires. In order to adjust it, the growth was monitored by reflection high-energy electron diffraction, which was used for fine tuning of the temperature over time in each growth run individually. For optimized growth conditions, a periodic diffraction pattern allowed for the detailed analysis of atomic arrangement on the surfaces and in the bulk. The ability to do so reflected the high crystal quality and ensemble uniformity of our CdTe NWs. The NW sides were formed by twelve stable, low-index crystalline facets. We observed two types stepped and polar sides, separated by in total six flat and non-polar facets.
The high crystalline quality of the cores allowed to grow epitaxial HgTe shells around. We reported on two different heterostructure geometries. In the first one, the CdTe NWs exhibit a closed HgTe shell, while for the second one, the CdTe NWs are overgrown mainly on one side. Scanning electron microscopy and scanning transmission electron microscopy confirmed, that many of the core-shell NWs are single crystalline zinc blende and have a high uniformity. The symmetry of the zinc blende unit cell was reduced by residual lattice strain. We used high-resolution X-ray diffraction to reveal the strain level caused by the small lattice mismatch in the heterostructures. Shear strain has been induced by the stepped hetero-interface, thereby stretching the lattice of the HgTe shell by 0.06 % along a direction oriented with an angle of 35 ° to the interface.
The different heterostructures obtained, were the base for further investigation of quasi-one-dimensional crystallites of HgTe. We therefore developed methods to reliably manipulate, align, localize and contact individual NWs, in order to characterize the charge transport in our samples. Bare CdTe cores were insulating, while the HgTe shells were conducting. At low temperature we found the mean free path of charge carriers to be smaller, but the phase coherence length to be larger than the sample size of several hundred nanometers. We observed universal conductance fluctuations and therefore drew the conclusion, that the trajectories of charge carriers are defined by elastic backscattering at randomly distributed scattering sites. When contacted with superconducting leads, we saw induced superconductivity, multiple Andreev reflections and the associated excess current. Thus, we achieved HgTe/superconductor interfaces with high interfacial transparency.
In addition, we reported on the appearance of peaks in differential resistance at Delta/e for HgTe-NW/superconductor and 2*Delta/e for superconductor/HgTe-NW/superconductor junctions, which is possibly related to unconventional pairing at the HgTe/superconductor interface. We noticed that the great advantage of our self-organized growth is the possibility to employ the metallic droplet, formerly seeding the NW growth, as a superconducting contact. The insulating wire cores with a metallic droplet at the tip have been overgrown with HgTe in a fully in-situ process. A very high interface quality was achieved in this case.
The combination of a topological insulator (TI) and a superconductor (S), which together
form a TI/S interface, is expected to influence the possible surface states in the
TI. It is of special interest, if the theoretical prediction of zero energy Majorana states
in this system is verifiable. This thesis presents the experimental realization of such
an interface between the TI strained bulk HgTe and the S Nb and studies if the afore
mentioned expectations are met.
As these types of interfaces were produced for the first time the initial step was
to develop a new lithographic process. Optimization of the S deposition technique as
well as the application of cleaning processes allowed for reproducible fabrication of
structures. In parallel the measurement setup was upgraded to be able to execute the
sensitive measurements at low energy. Furthermore several filters have been implemented
into the system to reduce high frequency noise and the magnetic field control
unit was additionally replaced to achieve the needed resolution in the μT range.
Two kinds of basic geometries have been studied: Josephson junctions (JJs) and
superconducting quantum interference devices (SQUIDs). A JJ consists of two Nb contacts
with a small separation on a HgTe layer. These S/TI/S junctions are one of the
most basic structures possible and are studied via transport measurements. The transport
through this geometry is strongly influenced by the behavior at the two S/TI
interfaces. In voltage dependent differential resistance measurements it was possible
to detect multiple Andreev reflections in the JJ, indicating that electrons and holes are
able to traverse the HgTe gap between both interfaces multiple times while keeping
phase coherence. Additionally using BTK theory it was possible to extract the interface
transparency of several junctions. This allowed iterative optimization for the highest
transparency via lithographic improvements at these interfaces. The increased transparency
and thus the increased coupling of the Nb’s superconductivity to the HgTe
results in a deeper penetration of the induced superconductivity into the HgTe. Due
to this strong coupling it was possible to enter the regime, where a supercurrent is
carried through the complete HgTe layer. For the first time the passing of an induced
supercurrent through strained bulk HgTe was achieved and thus opened the area for
detailed studies. The magnetic dependence of the supercurrent in the JJ was recorded,
which is also known as a Fraunhofer pattern. The periodicity of this pattern in magnetic
field compared to the JJ geometry allowed to conclude how the junction depends
on the phase difference between both superconducting contacts. Theoretical calculations
predicted a phase periodicity of 4p instead of 2p, if a TI is used as weak link
material between the contacts, due to the presence of Majorana modes. It could clearly
be shown that despite the usage of a TI the phase still was 2p periodic. By varying
further influencing factors, like number of modes and phase coherence length in the
junction, it might still be possible to reach the 4p regime with bound Majorana states
in the future. A good candidate for further experiments was found in capped HgTe
samples, but here the fabrication process still has to be developed to the same quality
as for the uncapped HgTe samples.
The second type of geometry studied in this thesis was a DC-SQUID, which consists
of two parallel JJs and can also be described as an interference device between two JJs.
The DC-SQUID devices were produced in two configurations: The symmetric SQUID,
where both JJs were identical, and the asymmetric SQUID, where one JJ was not linear,
but instead has a 90° bent. These configurations allow to test, if the predicted
uniformity of the superconducting band gap for induced superconductivity in a TI
is valid. While the phase of the symmetric SQUID is not influenced by the shape of
the band gap, the asymmetric SQUID would be in phase with the symmetric SQUID
in case of an uniform band gap and out of phase if p- or d-wave superconductivity
is dominating the transport, due to the 90° junction. As both devices are measured
one after another, the problem of drift in the coil used to create the magnetic field has
to be overcome in order to decide if the oscillations of both types of SQUIDs are in
phase. With an oscillation period of 0.5 mT and a drift rate in the range of 5.5 μT/h
the measurements on both configurations have to be conducted in a few hours. Only
then the total shift is small enough to compare them with each other. For this to be
possible a novel measurement system based on a real time micro controller was programmed,
which allows a much faster extraction of the critical current of a device. The
measurement times were reduced from days to hours, circumventing the drift problems
and enabling the wanted comparison. After the final system optimizations it has
been shown that the comparison should now be possible. Initial measurements with
the old system hinted that both types of SQUIDs are in phase and thus the expected
uniform band gap is more likely. With all needed optimizations in place it is now up
to the successors of this project to conclusively prove this last point.
This thesis has proven that it is possible to induce superconductivity in strained
bulk HgTe. It has thus realized the most basic sample geometry proposed by Fu and
Kane in 2008 for the appearance of Majorana bound states. Based on this work it is
now possible to further explore induced superconductivity in strained bulk HgTe to
finally reach a regime, where the Majorana states are both stable and detectable.
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.
This doctoral thesis investigates magneto-optical properties of mercury telluride layers grown tensile strained on cadmium telluride substrates. Here, layer thicknesses start above the usual quantum well thickness of about 20 nm and have a upper boundary around 100 nm due to lattice relaxation effects. This kind of layer system has been attributed to the material class of three-dimensional topological insulators in numerous publications. This class stands out due to intrinsic boundary states which cross the energetic band gap of the layer's bulk.
In order to investigate the band structure properties in a narrow region around the Fermi edge, including possible boundary states, the method of highly precise time-domain Terahertz polarimetry is used. In the beginning, the state of the art of Teraherz technology at the start of this project is discussed, moving on to a detailed description and characterization of the self-built measurement setup. Typical standard deviation of a polarization rotation or ellipticity measurement are on the order of 10 to 100 millidegrees, according to the transmission strength through investigated samples. A range of polarization spectra, depending on external magnetic fields up to 10 Tesla, can be extracted from the time-domain signal via Fourier transformation.
The identification of the actual band structure is done by modeling possible band structures by means of the envelope function approximation within the framework of the k·p method. First the bands are calculated based on well-established model parameters and from them the possible optical transitions and expected ellipticity spectra, all depending on external magnetic fields and the layer's charge carrier concentration. By comparing expected with measured spectra, the validity of k·p models with varying depths of detail is analyzed throughout this thesis. The rich information encoded in the ellipitcity spectra delivers key information for the attribution of single optical transitions, which are not part of pure absorption spectroscopy. For example, the sign of the ellipticity signals is linked to the mix of Landau levels which contribute to an optical transition, which shows direct evidence for bulk inversion asymmetry effects in the measured spectra.
Throughout the thesis, the results are compared repeatedly with existing publications on the topic. It is shown that the models used there are often insufficient or, in worst case, plainly incorrect. Wherever meaningful and possible without greater detours, the differences to the conclusions that can be drawn from the k·p model are discussed.
The analysis ends with a detailed look on remaining differences between model and measurement. It contains the quality of model parameters as well as different approaches to integrate electrostatic potentials that exist in the structures into the model.
An outlook on possible future developments of the mercury cadmium telluride layer systems, as well as the application of the methods shown here onto further research questions concludes the thesis.
Novel appraches to the molecular beam epitaxy of core-shell nanowires in the group II telluride material system were explored in this work. Significant advances in growth spurred the development of a flexible and reliable platform for a charge transport characterization of the topological insulator HgTe in a tubular nanowire geometry. The transport results presented provide an important basis for the design of future studies that strive for the experimental realization of topological charge transport in the quantum wire limit.
Within the scope of this thesis two main topics have been investigated: the examination of micromagnetic sensors and transport of massive and massless Dirac fermions in HgTe quantum wells. For the investigation of localized, inhomogeneous magnetic fields, the fabrication and characterization of two different non-invasive and ultra sensitive sensors has been established at the chair ”Experimentelle Physik” of the University of Würzburg. The first sensor is based on the young technique named micro-Hall magnetometry. The necessary semiconductor devices (Hall cross structures) were fabricated by high-resolution electron beam lithography based on two different two dimensional electron gases (2DEGs), namely InAs/(Al,Ga)Sb- and HgTe/(Hg,Cd)Te- heterostructures. The characteristics have been examined in two different ways. Measurements in homogeneous magnetic fields served for characterization of the sensors, whereas the investigation of artificially produced sub-µm magnets substantiates the suitability of the devices for the study of novel nanoscale magnetic materials (e.g. nanowires). Systematic experiments with various magnets are in accordance with the theory of single-domain particles and anisotropic behavior due to shapes with high aspect ratio. The highest sensitivity for strongly localized fields was obtained at T = 4.2 K for a (200x200) nm^2 Hall cross - made from shallow, high mobility HgTe 2DEG. Although the field resolution was merely δB ≈ 100 µT, the nanoscale sensor size yields an outstanding flux resolution of δΦ = 2 10^(−3) Φ0, where Φ0 = h/2e is the flux quantum. Translating this result in terms of magnetic moment, the sensitivity allows for the detection of magnetization changes of a particle centered on top of the sensor as low as δM ≈ 10^2 µB, with the magnetic moment of a single electron µB, the Bohr magneton. The further examination of a permalloy nanomagnet with a cross-section of (100x20) nm^2 confirms the expected resolution ability, extracted from the noise of the sensor. The observed high signal-to-noise ratio validates the detection limit of this sensor in terms of geometry. This would be reached for a magnet (same material) with quadratic cross-section for an edge length of 3.3 nm. Moreover, the feasibility of this sensor for operation in a wide temperature range (T = mK... > 200 K) and high magnetic fields has been confirmed. The second micromagnetic sensor is the micro-SQUID (micro-Superconducting-QUantum-Interference-Device) based on niobium. The typical sensor area of the devices built in this work was (1.0x1.0) µm^2, with constrictions of about 20 nm. The characterization of this device demonstrates an amazing field sensitivity (regarding its size) of δB < 1 µT. Even though the sensor was 25 times larger than the best micro-Hall sensor, it provided an excellent flux resolution in the order of δΦ ≈ 5 10^(−4) Φ0 and a similar magnetic moment resolution of δM ≈ 10^2 µB. Furthermore, the introduction of an ellipsoidal permalloy magnet (axes: 200 nm and 400 nm, thickness 30 nm) substantiates the suitability for the detection of minuscule, localized magnetic fields. The second part of the thesis deals with the peculiar transport properties of HgTe quantum wells. These rely on the linear contribution to the band structure inherent to the heterostructure. Therefore the system can be described by an effective Dirac Hamiltonian, whose Dirac mass is tunable by the variation of the quantum well thickness. By fabrication and characterization of a systematical series of substrates, a system with vanishing Dirac mass (zero energy gap) has been confirmed. This heterostructure therefore resembles graphene (a monolayer of graphite), with the difference of exhibiting only one valley in the energy dispersion of the Brillouin zone. Thus parasitical intervalley scattering cannot occur. The existence of this system has been proven by the agreement of theoretical predictions, based on widely accepted band structure calculations with the experiment (Landau level dispersion, conductivity). Furthermore, another particularity of the band structure - the transition from linear to parabolic character - has been illustrated by the widths of the plateaus in the quantum Hall effect. Finally, the transport of ”massive” Dirac fermions (with finite Dirac mass) is investigated. In particular the describing Dirac Hamiltonian induces weak localization effects depending on the Dirac mass. This mechanism has not been observed to date, and survives in higher temperatures compared to typical localization mechanisms.
In the present thesis the MBE growth and sample characterization of HgTe structures is investigated
and discussed. Due to the first experimental discovery of the quantum Spin Hall effect
(QSHE) in HgTe quantum wells, this material system attains a huge interest in the spintronics
society. Because of the long history of growing Hg-based heterostructures here at the Experimentelle
Physik III in Würzburg, there are very good requirements to analyze this material
system more precisely and in new directions. Since in former days only doped HgTe quantum
wells were grown, this thesis deals with the MBE growth in the (001) direction of undoped
HgTe quantum wells, surface located quantum wells and three dimensional bulk layers. All
Hg-based layers were grown on CdTe substrates which generate strain in the layer stack and
provide therefore new physical effects. In the same time, the (001) CdTe growth was investigated
on n-doped (001) GaAs:Si because the Japanese supplier of CdTe substrates had a
supply bottleneck due to the Tohoku earthquake and its aftermath in 2011.
After a short introduction of the material system, the experimental techniques were demonstrated
and explained explicitly. After that, the experimental part of this thesis is displayed.
So, the investigation of the (001) CdTe growth on (001) GaAs:Si is discussed in chapter 4.
Firstly, the surface preparation of GaAs:Si by oxide desorption is explored and analyzed.
Here, rapid thermal desorption of the GaAs oxide with following cool down in Zn atmosphere
provides the best results for the CdTe due to small holes at the surface, while e.g. an atomic
flat GaAs buffer deteriorates the CdTe growth quality. The following ZnTe layer supplies the
(001) growth direction of the CdTe and exhibits best end results of the CdTe for 30 seconds
growth time at a flux ratio of Zn/Te ~ 1/1.2. Without this ZnTe layer, CdTe will grow in the
(111) direction. However, the main investigation is here the optimization of the MBE growth
of CdTe. The substrate temperature, Cd/Te flux ratio and the growth time has to be adjusted
systematically. Therefore, a complex growth process is developed and established. This optimized
CdTe growth process results in a RMS roughness of around 2.5 nm and a FWHM value
of the HRXRD w-scan of 150 arcsec. Compared to the literature, there is no lower FWHM
value traceable for this growth direction. Furthermore, etch pit density measurements show
that the surface crystallinity is matchable with the commercial CdTe substrates (around 1x10^4
cm^(-2)). However, this whole process is not completely perfect and offers still room for improvements.
The growth of undoped HgTe quantum wells was also a new direction in research in contrast
to the previous n-doped grown HgTe quantum wells. Here in chapter 5, the goal of very low
carrier densities was achieved and therefore it is now possible to do transport experiments in
the n - and p - region by tuning the gate voltage. To achieve this high sample quality, very precise
growth of symmetric HgTe QWs and their HRXRD characterization is examined. Here,
the quantum well thickness can now determined accurate to under 0.3 nm. Furthermore, the transport analysis of different quantum well thicknesses shows that the carrier density and
mobility increase with rising HgTe layer thickness. However, it is found out that the band
gap of the HgTe QW closes indirectly at a thickness of 11.6 nm. This is caused by the tensile
strained growth on CdTe substrates. Moreover, surface quantum wells are studied. These
quantum wells exhibit no or a very thin HgCdTe cap. Though, oxidization and contamination
of the surface reduces here the carrier mobility immensely and a HgCdTe layer of around 5 nm
provides the pleasing results for transport experiments with superconductors connected to the
topological insulator [119]. A completely new achievement is the realization of MBE growth
of HgTe quantum wells on CdTe/GaAs:Si substrates. This is attended by the optimization of
the CdTe growth on GaAs:Si. It exposes that HgTe quantum wells grown in-situ on optimized
CdTe/GaAs:Si show very nice transport data with clear Hall plateaus, SdH oscillations, low
carrier densities and carrier mobilities up to 500 000 cm^2/Vs. Furthermore, a new oxide etching
process is developed and analyzed which should serve as an alternative to the standard
HCl process which generates volcano defects at some time. However, during the testing time
the result does not differ in Nomarski, HRXRD, AFM and transport measurements. Here,
long-time tests or etching and mounting in nitrogen atmosphere may provide new elaborate
results.
The main focus of this thesis is on the MBE growth and standard characterization of HgTe bulk
layers and is discussed in chapter 6. Due to the tensile strained growth on lattice mismatched
CdTe, HgTe bulk opens up a band gap of around 22 meV at the G-point and exhibits therefore
its topological surface states. The analysis of surface condition, roughness, crystalline quality,
carrier density and mobility via Nomarski, AFM, XPS, HRXRD and transport measurements
is therefore included in this work. Layer thickness dependence of carrier density and mobility
is identified for bulk layer grown directly on CdTe substrates. So, there is no clear correlation
visible between HgTe layer thickness and carrier density or mobility. So, the carrier density is
almost constant around 1x10^11 cm^(-2) at 0 V gate voltage. The carrier mobility of these bulk
samples however scatters between 5 000 and 60 000 cm^2/Vs almost randomly. Further experiments
should be made for a clearer understanding and therefore the avoidance of unusable
bad samples.But, other topological insulator materials show much higher carrier densities and
lower mobility values. For example, Bi2Se3 exhibits just density values around 1019 cm^(-2)
and mobility values clearly below 5000 cm2/Vs. The carrier density however depends much
on lithography and surface treatment after growth. Furthermore, the relaxation behavior and
critical thickness of HgTe grown on CdTe is determined and is in very good agreement with
theoretical prediction (d_c = 155 nm). The embedding of the HgTe bulk layer between HgCdTe
layers created a further huge improvement. Similar to the quantum well structures the carrier
mobility increases immensely while the carrier density levels at around 1x10^11 cm^(-2) at 0
V gate voltage as well. Additionally, the relaxation behavior and critical thickness of these
barrier layers has to be determined. HgCdTe grown on commercial CdTe shows a behavior as
predicted except the critical thickness which is slightly higher than expected (d_c = 850 nm).
Otherwise, the relaxation of HgCdTe grown on CdTe/GaAs:Si occurs in two parts. The layer
is fully strained up to 250 nm. Between 250 nm and 725 nm the HgCdTe film starts to relax
randomly up to 10 %. The relaxation behavior for thicknesses larger than 725 nm occurs than
linearly to the inverse layer thickness. A explanation is given due to rough interface conditions
and crystalline defects of the CdTe/GaAs:Si compared to the commercial CdTe substrate. HRXRD and AFM data support this statement. Another point is that the HgCdTe barriers protect the active HgTe layer and because of the high carrier mobilities the Hall measurements provide new transport data which have to be interpreted more in detail in the future. In addition, HgTe bulk samples show very interesting transport data by gating the sample from the top and the back. It is now possible to manipulate the carrier densities of the top and bottom surface states almost separately. The back gate consisting of the n-doped GaAs substrate and the thick insulating CdTe buffer can tune the carrier density for Delta(n) ~ 3x10^11 cm^(-2). This is sufficient to tune the Fermi energy from the p-type into the n-type region [138].
In this thesis it is shown that strained HgTe bulk layers exhibit superior transport data by embedding between HgCdTe barrier layers. The n-doped GaAs can here serve as a back gate.
Furthermore, MBE growth of high crystalline, undoped HgTe quantum wells shows also new
and extended transport output. Finally, it is notable that due to the investigated CdTe growth
on GaAs the Hg-based heterostructure MBE growth is partially independent from commercial
suppliers.
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.