Quantum Oscillation Measurements on the Organic
Superconductors κ-(ET)2 Cu(NCS)2 and α-(ET)2 NH4
in High Magnetic Fields and Ultra–Low Temperatures
Thesis submitted to the faculty of Clark University, Massachusetts, in partial
fulfillment of the requirements for a Master’s of Arts degree in the
Department of Physics
by
Alireza Alipour
August 2022
Thesis Committee:
Prof. Charles Agosta, Ph.D.
Chief Instructor
Prof. Ranjan Mukhopadhyay, Ph.D.
Professor of Physics
Abstract
The field of Fermiology has evolved into an established and necessary technique for
investigating the electronic topology of organic conductors in solid–state physics. Organic conductors have recently gained considerable attention due to their various
electronic states such as the Mott insulating state, superconducting state, metallic
state, and charge ordering state. Due to their layered structures consisting of organic molecules and anions, most organic conductors exhibit highly two–dimensional
(2D) electronic states. At low temperatures, these layered organic conductors show
intriguing physical properties based on their spin, charge, and orbital degrees of freedom. Strong electron correlations in electronic structures play an essential role in
exhibiting such properties. Hence, the study of strong correlation effects in electronic
states with organic conductors has become an important target, alongside studies of
rare–earth metal complexes and transition metal oxides.
Understanding the superconductivity mechanism in organic metals requires an
in–depth study of the electronic band structure, especially the shape and size of the
Fermi surface. Experimental measurements of Fermi surfaces are possible using magnetic quantum oscillations, such as de Haas–van Alphen (dHvA), Shubnikov–de Haas
(SdH), and angular dependent magnetoresistance oscillations (AMRO). Frequencies
and amplitudes of quantum oscillations vary with external parameters, such as magnetic field, temperature, and crystal orientation towards the magnetic field. Among
the organic superconductors, organic charge–transfer salts (ET)2 X, where ET is the
donor layer and X stands for monovalent anions, are considered the most promising due to their layered crystal structure, 2D electronic structure, and low critical
temperature superconductivity at ambient pressure.
In this thesis, we studied the Shubnikov–de Haas (SdH) quantum oscillations of
the two organic superconductors, κ–(ET)2 Cu(NCS)2 and α–(ET)2 NH4 in the normal
state. The magnetoresistance of these superconductors was obtained by using the
tunnel diode oscillator (TDO) technique under very high magnetic fields (30 T), very
low temperatures (33 mK), and a wide range of angles (up to 50°). We applied the
Filter–Peak–Fit (FPF) method to extract the frequency and amplitude variations in
the SdH quantum oscillations and then measured the normal state parameters like
the effective mass m∗ , the Dingle temperature TD , and the Spin splitting factor (g–
factor) using the Lifshitz–Kosevich equations. Information about the many–body
interactions and the crystal purity can be obtained from these measurements for
the 2D quasi–organic superconductors. Field–, temperature–, and angel–dependent
oscillations are analyzed and find some anomalous features, e.g., a field and orbital dependence of the effective mass, non–linearity of the Dingle plots, and orbit–dependent
Dingle temperature parameter. As a result, These differences appear to depart from
Lifshitz–Kosevich’s theory for the materials due to their highly two–dimensional nature.
© 2022 Alireza Alipour
All Rights Reserved.
Academic History
Name (in full): Alireza Alipour, Date: August 2022
Baccalaureate Degree: B.S. Physics, Date: June 2011
Source: University of Guilan, Rasht, Guilan, Iran
Other Degrees, with dates and sources:
Master Degree: M.S. Physics, Date: September 2014
Source: Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran
Occupation and Academic Connection:
Math and Physics Teacher, Ghazali High School, Tehran, Iran 2014-2018
Math Teacher, Borna High School, Tehran, Iran 2018-2019
Physics and Math Teacher, Homa High School, Tehran, Iran 2019-2020
Clark University Graduate School 2020-2022
Teaching and Research Assistant, Clark University 2020-2022
Physics Teacher, Wachusett Regional High School, Holden, MA March 2022-June 2022
Dedication
I dedicate my thesis work to my family. A special feeling of gratitude to my loving
parents whose words of encouragement and push for tenacity ring in my ears. My
sisters and brothers have never left my side and are very special. I am truly thankful
for having you all in my life.
vi
Acknowledgements
Throughout the writing of this dissertation, I have received a great deal of support
and assistance. I would first like to thank my supervisor, Professor Charles C. Agosta,
whose expertise was invaluable in formulating the research questions and methodology. Your insightful feedback pushed me to sharpen my thinking and brought my
work to a higher level. I would like to acknowledge my colleagues for their patient
support and for all of the opportunities I was given to further my research. Finally, I
would like to thank my parents for their wise counsel and sympathetic ear. You are
always there for me.
vii
Contents
List of Tables
x
List of Figures
xi
1 General Introduction and Purpose
1.1
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Discovery of Superconductivity . . . . . . . . . . . . . . . . .
2
1.1.2
Organic Superconductors . . . . . . . . . . . . . . . . . . . . .
7
1.1.3
Shubnikov–de Hass Oscillations Measurements . . . . . . . . .
11
1.2
The purpose of this thesis . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2 Properties of Superconductors κ-(ET)2 Cu(NCS)2 and α-(ET)2 NH4 16
2.1
Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3 Theory of Magnetic Quantum Oscillations
viii
28
3.1
3.2
The Origin of Quantum Oscillations . . . . . . . . . . . . . . . . . . .
30
3.1.1
Landau Quantization . . . . . . . . . . . . . . . . . . . . . . .
30
3.1.2
Quantum Oscillation Frequency . . . . . . . . . . . . . . . . .
31
Lifshitz–Kosevich (LK) Equation . . . . . . . . . . . . . . . . . . . .
33
3.2.1
De Haas–van Alphen (dHvA) Oscillations . . . . . . . . . . .
34
3.2.2
Shubnikov–de Hass (SdH) Oscillations . . . . . . . . . . . . .
36
3.2.3
Damping Terms . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4 Data Analysis and Results
4.1
44
Damping Factors Calculation . . . . . . . . . . . . . . . . . . . . . .
45
4.1.1
Filter–Peaks–Fit (FPF) Method . . . . . . . . . . . . . . . . .
46
4.1.2
Effective Mass Calculation . . . . . . . . . . . . . . . . . . . .
50
4.1.3
Calculation of the Dingle Temperature . . . . . . . . . . . . .
56
5 Conclusions
60
ix
List of Tables
2.1
The room temperature crystallographic data and the . . . . . . . . .
21
4.1
Effective masses determined for the superconductor . . . . . . . . . .
51
4.2
Dingle temperatures determined for the superconductor . . . . . . . .
57
x
List of Figures
1.1
Heike Kamerlingh Onnes at his liquefactor, Leiden, the . . . . . . . .
2
1.2
(a) Periodic table of superconducting elements. Taken . . . . . . . . .
3
1.3
A diamond anvil creates a superconductor at room . . . . . . . . . .
6
1.4
Molecular structures of (a) tetramethyltetraselenafulvalene . . . . . .
8
1.5
The conducting layers including BEDT–TTF molecules . . . . . . . .
9
1.6
(a) The Shubnikov–de Haas oscillations observed in the . . . . . . . .
12
1.7
(a) The Shubnikov–de Haas oscillations observed in the . . . . . . . .
13
2.1
(a) The κ–(BEDT − TTF)2 X crystal structure. (b) The . . . . . . . .
20
2.2
The κ–(ET)2 Cu(NCS)2 unit cell. (a) The κ–(ET)2 Cu(NCS)2 . . . . .
21
2.3
The α–(ET)2 NH4 Hg(SCN)4 crystal structure projected . . . . . . . .
22
2.4
Illustration of orbital hybridization in a schematic form . . . . . . . .
24
2.5
(a) A calculation of the band structure and the Fermi . . . . . . . . .
26
3.1
An illustration of the Landau tubes for (a) a 3D isotropic . . . . . . .
32
3.2
Computed RT (p) as a function of magnetic field and . . . . . . . . . .
39
3.3
The finite relaxation time effect. RD (p) as a function of . . . . . . . .
40
xi
3.4
Schematic illustrations of (a) the Zeeman splitting of the . . . . . . .
41
4.1
Illustration of the Filter–Peaks–Fit (FPF) method . . . . . . . . . . .
47
4.2
Magnetoresistance versus magnetic field. The inset shows . . . . . . .
48
4.3
(a) Raw oscillations in magnetoresistance versus inverse . . . . . . . .
49
4.4
(a) The α oscillations and (b) the β oscillations vs inverse . . . . . .
50
4.5
Magnetoresistance oscillations as a function of temperature . . . . . .
53
4.6
(a) The α–amplitudes and (b) the 2α–amplitudes at different . . . . .
53
4.7
Amplitude of the QOs for (a) the α–orbit and (b) the 2α–orbit . . . .
54
4.8
The ln(RT /T ) versus temperature T diagram which gives . . . . . . .
55
4.9
Effective mass of α and 2α orbits as a function of field . . . . . . . .
56
4.10 The ln(RD ) vs. 1/B for different orbits in constant temperature . . .
58
4.11 The ln(RD ) vs. 1/B for different angels. From the fits, the . . . . . .
59
xii
Chapter 1
General Introduction and Purpose
1.1
Introduction
A slight warmth is probably felt if you put your hand on the power supply of your
laptop computer or another device. As your device converts household power into a
current, heat is an unwanted byproduct. Electric power transmission produces RI 2
line losses. No matter how the electricity is generated, whether it is generated by
nuclear plants, solar plants, hydroelectric plants, or wind farms, line losses exist.
Using a higher voltage can reduce these losses, but not eliminate them. These line
losses could be eliminated if transmission lines had no resistance, but that would
require highly conductive lines. An effort to reduce or eliminate this unwanted thermal
energy would be a significant accomplishment in a world where energy waste is a global
concern. This is possible by using superconductors.
The phenomenon of superconductivity is a physical property exhibited by certain
materials in which their electrical resistance disappears and magnetic flux fields are
emitted from certain materials. Superconductors are materials that exhibit these
properties. As the temperature is lowered, the resistance of an ordinary metallic
1
conductor decreases gradually while the resistance of a superconductor drops abruptly
to zero below a critical temperature. It is possible to sustain an electric current
through a loop of superconducting wire indefinitely without power. In this chapter,
we present a brief history of superconductors and review organic superconductors
and Shubnikov–de Hass (SdH) oscillations measurements. Finally, the purpose of
this thesis and its structure are introduced.
1.1.1
Discovery of Superconductivity
Heike Kamerlingh Onnes (see Figure 1.1a) of Leiden University, a Dutch Nobel Laureate, investigated the temperature dependence of mercury’s resistance in 1911 [1].
As the mercury sample was cooled to 4.2 K (−269.2 °C) using liquid helium, it shows
zero electrical resistance. For mercury, this temperature is defined as a critical temperature Tc (see Figure 1.1b). The mercury sample reached a phase in which the
resistance was zero. Such a state is known as superconductivity. At very low temperb
a
Figure 1.1: (a) Heike Kamerlingh Onnes at his liquefactor, Leiden, the Netherlands, 1913. Adapted
from Ref. [2]. (b) Onnes’ measurements of resistance versus temperature in mercury, showing the
superconductivity below 4.2 K [1].
2
atures, mercury exhibits zero resistance–It is a superconductor up to approximately
4.2 K. When it reaches that critical temperature, its resistance changes dramatically
and then increases linearly as temperature increases. Since the first observation of
superconductivity in 1911, the transition temperature Tc has gradually increased.
Interestingly, copper, silver, and gold, which produce high conductivities, have no superconducting properties (see Figure 1.2a). For many years, the critical temperature
Tc of superconductors was a few degrees Kelvin. However, the Tc is still extremely
a
b
Figure 1.2: (a) Periodic table of superconducting elements. Taken from Ref. [3]. (b) Discovery
time of critical temperatures Tc . Since 1986, when the first high–Tc superconductor was observed,
the critical temperature of superconductors has increased sharply. Taken from Ref. [4].
3
cold and close to liquid helium temperatures. The cooling requirements make it
uneconomical to transmit electric energy at this temperature. The search for high–
temperature superconductors and understanding the mechanism of such a behavior
has been one of the hot spots in condensed matter physics ever since Onnes discovered
the phenomenon of superconductivity in 1911.
Initially people searched superconductors mainly among the materials containing
transition metals with superconducting transition temperatures Tc = 17–23 K. It was
found in 1941 that the Niobium–Nitride (NbN) alloy was superconductor at Tc = 16
K (-257 °C), and in 1953, it was found that the Vanadium–Silicon (VSi2 ) became
superconductor at Tc = 17.5 K (-255.7 °C). The Niobium–Tin (Nb3 Sn)’s superconducting transition was detected at Tc = 18 K in 1954 [5] by measuring the inductance
of a coil encircling the sample. Due to the Meissner effect [6], the magnetic field
is expelled at the superconducting transition, thus reducing the inductance. In this
experiment, the coil was immersed in liquid hydrogen and the inductance measured
as a function of temperature. All this changed in 1986 in an IBM lab in Zurich,
when J. Georg Bednorz and K. Alex Müller [7] produced a ceramic material known
as La2−x Bax CuO4. It became superconducting with a higher critical threshold temperature of 35 K (-238 °C). This ceramic material was the first of a new class of
high–temperature superconductors. It is made by randomly substituting some Barium atoms into the lattice of Lanthanum–Copper–Oxide in what is termed a solid
solution. This discovery earned a Noble Prize in Physics in 1987 and led to another
breakthrough only a year later. A large jump was seen in 1986 (see Figure 1.2b),
when researchers from Houston University, led by Dr. Ching Wu Chu [8], synthesized
a brittle, ceramic compound with a transition temperature of Tc = 92 K (-181 °C).
Even though this temperature is still quite cold, it is close to the boiling point of liquid
nitrogen (77 K). At room temperature, this ceramic material, made of Yttrium Bar4
ium Copper Oxide (YBa2 Cu3 O7 ), acted as an insulator. In contrast to conventional
superconductors, the high–Tc superconductors exhibit very unusual properties. First
of all, electron pairs are not likely to be coupled in a conventional electron–phonon
interaction exhibiting the s–wave symmetry. Rather, the pairing mechanism proposes
a d–wave symmetry [9, 10]. Secondly, There is no superconducting gap as the temperature approaches the critical point. In its normal state, however, the energy gap,
commonly known as the pseudo–gap, is observed [10, 11]. From scanning tunneling
microscopy and angle–resolved photoemission spectroscopy, it has been observed that
two energy gaps coexist [12]. The most interesting thing about it is that there is a
2D Fermi surface which is thought to contribute to the high critical temperature.
The observation of room–temperature superconductivity is one of the interesting challenges in experimental condensed matter physics. Over the past decade,
researchers have been investigating copper oxides and iron–based chemicals as high–
temperature superconductors. Hydrogen has, however, also been identified as a potentially good building block in the universe (the most abundant element). Scientists
are compressing molecular solids with hydrogen to achieve a new record for materials that are superconductive at room temperatures. Several systems under high
pressure have been reported to exhibit conventional superconductivity at high temperatures in hydrogen–rich materials [13, 14, 15]. A significant discovery that led to
room–temperature superconductivity with a transition temperature of 203 K was the
disproportionation of hydrogen sulfide (H2 S) to H3 S as a result of pressures exceeding
155 GPa [13, 16].
Despite decades of speculation, these works validate physics predictions made
decades ago by Neil Ashcroft at Cornell University in Ithaca, New York, that materials rich in hydrogen would superconduct at much higher temperatures than previously
thought. The most fundamental requirements for a high–temperature superconductor
5
Figure 1.3: A diamond anvil creates a room–temperature superconductor. Taken from Ref. [17, 18].
are stronger bonds and light elements. Hydrogen is the lightest element, and in terms
of strength, the hydrogen bond is the strongest. According to theory, solid metallic
hydrogen has a high Debye temperature and strong electron–phonon coupling that
gives rise to room–temperature superconductivity [19]. However, pure hydrogen gets
into a metallic state at extremely high pressures, which was achieved for the first time
in a lab in 2017 by Isaac Silvera and Dias [19]. Furthermore, a new methodology was
developed for synthesizing superconducting materials at lower pressures [19] (see Figure 1.3). A diamond anvil cell, which is used to test minuscule amounts of chemicals
under exceptionally high pressure, was used in their research to photochemically synthesize organic–derived carbonaceous sulfur hydride by combining hydrogen, carbon,
and sulfur. The resulting material is superconducting at a temperature of about 287.7
K and a pressure of about 267 GPa [17]. In addition, a thin palladium film was used
to separate hydrogen atoms from yttrium resulting in yttrium super–hydride which
exhibited superconductivity at 262 K and about 182 GPa [18].
6
1.1.2
Organic Superconductors
Typically, organic materials are considered to be electrical insulators. In contrast,
there are some organic compounds that have metallic conductivity at room temperature and usually, their conductivities are lower than Copper. One of the most significant developments in the field of organic metals was the discovery of TTF–TCNQ [20,
21], which consists of the organic electron acceptor tetracyano–p–quinodimethane
(TCNQ) and the electron donor tetrathiafulvalene (TTF). Metallic conductivity, σ,
is observed with σ ≈ 104 Scm−1 near 60 K along the stacking direction for this
material (The σ is about 106 Scm−1 for Copper at room temperature) [22].
A major goal of the investigations into organic metals is to pursue superconductivity. In 1964, Little [23] proposed synthesizing an organic polymeric superconductor
for the first time, as well as considering the effect of dimensionality in superconductivity, then Ginzburg [24] applied these ideas to two–dimensional systems. Introducing
the donor TTF by Wudl [25] in 1970, as well as Shirakawa’s discovery [26] of the
properties of doped polyacetylene were two significant achievements in the period
leading up to the discovery of the first organic superconductor. The most significant
finding was the discovery of superconductivity in a series of quasi–one–dimensional
organic conductors TMTSF (tetramethyltetraselenafulvalene), followed by a number
of quasi–two–dimensional BEDT–TTF (bisethylenedithio–tetrathiafulvalene or ET
for short). Figure 1.4 depicts the basic molecules in two different types of organic
superconductors. The BEDT–TTF molecule is a major component of many superconductors [22]. The materials were synthesized based on Little’s hypothesis [23] in
which electrons could easily pair with Cooper pairs in organic polymers that have
highly polarizable side chains. Although there is no proof that the ET materials support this mechanism, this leads to a very high temperature at which superconducting
7
a
b
Figure 1.4: Molecular structures of (a) tetramethyltetraselenafulvalene (TMTSF) and (b)
bisethylenedithio–tetrathiafulvalene (BEDT–TTF or ET for short). Adapted from Ref. [22, 27].
transitions occur. This is due to the area of positive charge responsible for the attractive interaction being caused by the displacement of an electron whose mass is many
times smaller than those of the moving ions that create a lattice vibration in inorganic
superconductors [22]. Organic superconductors were first discovered by Jerome and
colleagues [28] in 1980 with the Bechgaard salts [29] based on TMTSF, (TMTSF)2 X
(X= PF6 , ClO4 , etc.) [30]. It is only under extremely high pressure of ≈ 12 kbar that
(TMTSF)2 PF6 exhibits superconductivity with a critical temperature of Tc ≈ 0.9
K [22, 28]. At ambient pressure, (TMTSF)2 ClO4 shows superconductivity with a
Tc ≈ 1.4 K [22, 29]. In 1983, an IBM Almaden group [31] successfully synthesized
a new family of organic metals called (BEDT – TTF)2 X, or (ET)2 X for abbreviation,
and found one of the members, (ET)2 ReO4 , to be superconducting below ≈ 2.5 K.
8
For this compound to suppress an insulating state and become superconducting, a
pressure of 4.5 kbar is also required.
In addition to the fullerenes, the highest Tc for the organic charge–transfer salt
is also made from the ET molecule. The κ–(ET)2 Cu[N(CN)2 ]Cl has a complex low–
temperature phase diagram but indicates superconductivity at a transition temperature of Tc ≈ 12.5 K with only moderate pressure of 0.3 kbar [22, 32]. Although a
phase diagram for κ–(ET)2 Cu[N(CN)2 ]Cl at low temperature shows complex phases,
it exhibits superconductivity at pressures as low as 0.3 kbar at Tc ≈ 12.5 K. Unlike
the Bechgaard salts which are characterized by a quasi–1D structure, the ET materials have a two–dimensional electronic structure. The electronic bands in these salts
are created by overlapping π orbitals [22]. The distances between Se–Se and S–S is
smaller than the van der Waals radius of 3.96 Å or 3.6 Å, respectively, in certain
directions [22]. As of now, researchers have developed more than 50 kinds of superconductors containing conducting BEDT–TTF layers alternating with insulating
anion layers, resulting in a quasi–two–dimensional structure [33]. A characteristic of
BEDT–TTF salts is that there are a variety of arrangements among the BEDT–TTF
layers. Greek letters such as α, β, κ, and λ are used to identify the packing arrangement of BEDT–TTF molecules, as illustrated in Figure 1.5. Superconductivity can be
explored in a variety of ways since each arrangement represents a different electronic
system [34].
Figure 1.5: The conducting layers including BEDT–TTF molecules with different packing arrangements. Taken from Ref. [34].
9
Fermi surfaces (FS) play an important role in understanding metallic and superconducting behavior. The organic conductors are unusually pure systems that display
unusually strong magnetotransport signatures such as quantum oscillations as a result of their FS. These materials possess important physical properties due to their
electronic structures, or FS. The FS topologies were theoretically studied using the
tight–binding approximation, two–dimensional Hubbard model, and first principle
self–consistent local–density method.
Measurements of the de Haas–van Alphen (dHvA) or magnetization, Shubnikov–
de Haas (SdH) or magnetoresistance effect, and angular dependent magnetoresistance
oscillations (AMRO) are crucial to determine the topology of the FS [34]. Charge
carriers in ⃗k space must be in a closed extremal orbit in order to observe these
magnetic quantum oscillations [22]. As a consequence, open FS sheets in 1D materials
cannot show the dHvA or SdH oscillations. In contrast, 2D organic conductors are the
ideal materials to investigate both dHvA and SdH effects [22]. As the sample quality
improved, the first SdH oscillations in κ–(ET)2 Cu(NCS)2 [35] and β–(ET)2 IBr2 [36,
37] were discovered in 1988. Magnetic quantum oscillations were discovered in many
other organic metals following this discovery [22, 38].
Understanding the electronic properties of organic charge–transfer salts was greatly
enhanced by these studies. The exact FS topology of many compounds could be
mapped out, allowing direct comparison with the band–structure predictions. The
experimental and calculated FS often agreed remarkably well [22]. Calculations of
the FS were made using the tight–binding Huckel method [39, 40, 41] based on some
approximations. The ET compounds have a 2D band structure that leads to an FS
of nearly cylindrical shape. Organic metals are three–dimensional crystals. Their
electronic structure also exhibits a certain degree of three–dimensionality. As a result, a striped form of the FS is generated, leading to a unique form of AMRO. In
10
light of this, measuring the angular dependence of the resistance in magnetic fields
has emerged as an important tool for providing information about band structures in
low–dimensional systems [22].
1.1.3
Shubnikov–de Hass Oscillations Measurements
Magnetic quantum oscillations measurements conducted on organic metals have yielded
new insights into the nature of low–temperature states [22]. Under pressure, the SdH
measurements indicate the changing band–structure caused by the increasing dimensionality. The dHvA oscillations in the superconducting state have only been observed
in one ET salt [22, 42, 43] and come with more clues about the scattering mechanism
of the quasi–particles in the Shubnikov phase.
The effective cyclotron mass can be determined by analyzing the temperature dependence of dHvA (or SdH) oscillations. There are sometimes discrepancies between
the mass measured by these methods and that obtained by the band–structure calculations, cyclotron resonance, and specific heat measurements [22]. There is still
considerable debate over whether electron–electron or electron–phonon interactions
are responsible for this discrepancy, and further studies are required to determine the
answer.
The Shubnikov–de Haas (SdH) oscillations give information about the Fermi surface (FS) based on a linear relationship between the frequency of the SdH oscillations
and the extremal area of the FS. For the first time, Oshima et al. [35] confirmed experimentally the Fermi surface of the organic superconductor κ–(ET)2 Cu(NCS)2 by
performing magnetoresistance measurements at ambient pressure. The Shubnikov–
de Haas (SdH) oscillations was observed by changing crystal orientation up to 30°
at temperatures less than 1 K and magnetic fields above 8 T (see figure 1.6a). The
11
a
b
Figure 1.6: (a) The Shubnikov–de Haas oscillations observed in the superconductor κ–
(ET)2 Cu(NCS)2 below 1 K. (b) The Fermi Surface of the κ–(ET)2 Cu(NCS)2 including closed hole–
like quasi–2D orbit (α pockets) and two open quasi 1D corrugated sheets, separated by an energy
gap, as determined by high magnetic field measurements. Taken from Ref. [35].
cylindrical closed FS surface of the κ–(ET)2 Cu(NCS)2 consists of the closed hole–
like orbit (α orbit) which has an area of approximately 18% of the area of the first
Brillouin zone and obtained from the frequency of the SdH oscillations (625 T).
Later, new SdH oscillations with frequency of 3800 T has been observed for κ–
(ET)2 Cu(NCS)2 at fields up to 24 T by Sasaki et al. [44]. At high magnetic fields,
through the magnetic tunnelling of the electrons through the gap between the hole–
like and electron–like orbits, larger orbits are possible. These orbits are called β orbits
(magnetic breakdown oscillations), and their area is approximately 100% of the area
of the first Brillouin zone. Figure 1.6b shows the Fermi surface of κ–(ET)2 Cu(NCS)2
consists of closed hole–like quasi–2D orbits (α pockets) and two open quasi 1D corrugated sheets, separated by an energy gap.
The Shubnikov–de Haas (SdH) measurements of the organic superconductor α–
(ET)2 NH4 were reported by Osada et al. [45] using pulsed high magnetic fields up to
35 T (see figure 1.7a). The large Shubnikovde Haas (SdH) oscillations and the angle–
12
a
b
Figure 1.7: (a) The Shubnikov–de Haas oscillations observed in the superconductor α–(ET)2 NH4
for several directions of applied fields at the fixed temperature 1.5 K. The angle dependence of the
resistance at the fixed field 6.4 T plotted in the inset. Adapted from Ref. [45]. (b) The Fermi Surface
of the α–(ET)2 NH4 including closed–orbit (around the BV point) and open–orbit (along the Γ Z
line). Taken from Ref. [46]
dependent quantum oscillations have been observed in the organic superconductor.
The strong two–dimensionality and the existence of a cylindrical Fermi surface with
weak warping has been suggested [45]. Figure 1.7b shows the Fermi surface of α–
(ET)2 NH4 consists of closed–orbit (around the BV point) and open–orbit (along the
Γ Z line).
Later, Sundhu et al. [47] reported the SdH measurements on the α–(ET)2 NH4 in
pulsed fields up to 50 T. The field and temperature–dependence of the oscillations
have been analyzed and showed several unusual features, such as a field–dependent
effective mass parameter, non–linear Dingle plots and an anomalously Large harmonic
ratio. These results suggest departures from the Lifshitz–Kosevich formalism, arising
from the highly two–dimensional nature of the superconductor [47].
13
1.2
The purpose of this thesis
In this thesis, an overview of the current knowledge of the Fermi surfaces and the
metallic state properties including crystal structures and the energy band structures
of the quasi–2D organic superconductors α–(ET)2 NH4 and κ–(ET)2 Cu(NCS)2 is presented based on magnetic quantum measurements of the Shubnikov–de Haas (SdH)
and angular dependent magnetoresistance oscillations (AMRO). These measurements
were made using a contactless method, called the tunnel diode oscillator (TDO) technique. In the TDO method, the sample is placed in the coil of a radio frequency (rf)
tank circuit oscillating and the in–plane resistivity is measured. We applied the Filter–
Peak–Fit (FPF) method to extract the frequency and amplitude variations in the SdH
quantum oscillations of the superconductors α–(ET)2 NH4 and κ–(ET)2 Cu(NCS)2 .
The quantum oscillation frequencies as well as their amplitudes depend on the values
of the external parameters such as magnetic field, temperature, and crystal orientation with respect to the magnetic field. By performing measurements at very high
magnetic fields (30 T), very low temperatures (33 mK), and a large range of angles
(up to 50°), three important band structure parameters such as the effective bare
band mass m∗ , the Dingle temperature TD and the spin splitting factor (g–factor) are
calculated for different orbits using the Lifshitz–Kosevich equations and are compared
with other measurements. Information about the many—body interactions and the
crystal purity can be obtained from these measurements for the 2D quasi–organic
superconductors. Field, temperature, and angel–dependent oscillations are analyzed
and find some anomalous features, e.g., a field and orbital dependence of the effective mass, non–linearity of the Dingle plots, and orbit–dependent Dingle temperature
parameter. As a result, These differences appear to depart from Lifshitz–Kosevich’s
theory for the materials due to their highly two–dimensional nature.
14
1.3
Thesis Structure
This thesis consists of five chapters. Following the introduction in Chapter 1 which
focuses mainly on historical background of superconductors, Chapter 2 summarizes
basic properties of organic superconductors α–(ET)2 NH4 and κ–(ET)2 Cu(NCS)2 , including their synthesis, structure, and electronic properties. An overview of the theory
of magnetic quantum oscillations and angular–dependent magnetoresistance oscillations is presented in Chapter 3. A detailed discussion of the experimental results
can be found in Chapter 4. Chapter 5 gives some conclusions based on the data
analysis. Finally, a detailed description of the implementation of the Filter–Peak–Fit
(FPF) method is given in Appendix ??.
A continuous stream of new results is coming out in this highly topical area of
the ”Magnetic Quantum Oscillation Measurements of the Organic Superconductors”.
Therefore, this study can only provide an overview of the current state of the subject.
Many of the present–day problems may be resolved in the next few years, while new
insights into solid–state properties may emerge.
15
Chapter 2
Properties of Superconductors
κ-(ET)2Cu(NCS)2 and α-(ET)2NH4
The organic donor molecule BEDT − TTF (or ET for short) has so far the highest critical temperature Tc in the most crystalline organic superconductors [48]. To demonstrate their wide variety of stoichiometry, ET salts can be indicated by (ET)m Xn .
Changing the valence of ET is a relatively simple process. Many superconducting
compounds, however, have the 2:1 structure (ET)2 X, where X is a monovalent anion [48].
The fermiology (Fermi surface topology) of the organic metals with cation radical
salts of the ET, has attracted a great deal of attention in connection with organic
superconductivity and the validity of band calculation in these materials [35]. In the
organic superconductors, there are several types in packed ET donor layers, such as
α–, β–, κ–, and λ–type arrangements (see Figure 1.5). The κ–phase organic conductor
of the donor molecule ET with the general formula κ–(ET)2 X has become one of
the most studied families of the organic conductors during recent years. A common
structural pattern to all of these materials is that they have highly conducting layers
16
of donor ET alternating with insulator layers of the anions in one crystallographic
direction. This results in the low dimensional character of their electronic properties.
While the carrier concentration amounts to half a hole per ET molecule in most of the
conducting ET compounds, the κ phase is characterized by a specific ET arrangement.
The superconductor κ–(ET)2 Cu(NCS)2 is one of the most intensively studied systems among the κ–(ET)2 X family of organic superconductors and exhibits one of the
highest Tc of 10.4 K at ambient pressure. The κ–(ET)2 Cu(NCS)2 ’s quasi–2D Fermi
surface is similar to Pippard’s textbook model [49] of a chain of coupled orbits. Because of this feature, it is ideal for examining interesting topics in fermiology, such as
the ”competing coexistence” of different types of quantum oscillations or the study
of phenomena such as a magnetic breakdown.
An isostructural series of compound α–(ET)2 MHg(SCN)4 (M = K, Rb, and
NH4 ) [35, 50, 51], has been prepared as a modification of an organic superconductor
κ–(ET)2 Cu(NCS)2 . The donor molecular conducting planes are sandwiched between
thick insulating anion layers of this system, resulting in a highly two–dimensional
electronic structure. A unique electronic structure is created due to the chevron–like
arrangement of donor molecules in the conducting plane. Although these compounds
have almost the same lattice parameters, only the NH4 salt is superconducting (Tc =
0.8 K) at ambient pressure [46]. It is still unclear what makes their electrical properties different. In this chapter, we present the chemical properties (i.e. crystal
structures) and Physical properties (i.e. Fermi surfaces) of the organic superconductors κ–Cu(NCS)2 and α–NH4 as determined from theoretical calculations and
experimental measurements.
17
2.1
Synthesis
The comprehensive description of the synthesis of various organic donor molecules and
of the experimental methods to generate organic conducting salts has been presented
in several articles [48, 52]. Electrocrystallization is the process used to make high–
quality single crystals by oxidizing the organic electron donor molecules [22, 48].
Using the same technique, charge–balancing anions are used to form crystals. For
crystal growth, platinum electrodes and a glass frit of ultrafine porosity are used
in an H–shaped cell. In the anode compartment, donor molecules, e.g., ET are
placed in a suitable solvent. In the cathode and anode compartments, a supporting
electrolyte, e.g., NBu4 X, is added, with Bu representing n–butyl and X representing
the monovalent anion, such as Cu(NCS)2 – . The electrocrystallization process takes a
while, ranging from a few days up to several months. Crystal growth occurs when the
current density is kept at a minimum (≈ 1 µ A/cm2 ) [22, 48]. In order to prevent the
formation of divalent donor ions and obtain large single crystals, the current density
is decreased after the generation of seed crystals. The quality of the crystals can be
improved by a second and third growth process in the same solution after the first
batch has been harvested. These quasi–1D materials are needle–like and black in color
and the ones that conduct the best have long axes of several millimeters. Salts with
a 2D structure have plates with a few millimeter side lengths and a thinner thickness
perpendicular to the highly conducting planes of a few tenths of a millimeter. It is
essential to use the best solvent, the correct current density, and the best starting
material to obtain the various materials based on ET donor molecules [22, 48].
18
2.2
Crystal Structure
Molecular conductors usually contain donor (cation) molecules and acceptor (anion) molecules. The (BEDT − TTF)2 X salt, for instance, has two BEDT − TTF
molecules giving an electron to the X anion (acceptor) molecule, thus creating one
hole per two BEDT − TTF molecules. Cation and anion radicals are formed by transferring charges between molecules during salt formation. Due to this charge transfer
property, molecular conductors are also referred to as charge transfer salts.
Low dimensionality is a key characteristic of organic conductors, primarily due to
the strong anisotropy of molecular structures. According to these examples, the π
conjugated systems can form when the large p orbitals from the carbon or sulfur atoms
of the donor molecules overlap. Anion molecules become insulating as a result of
charge transfer and generate insulating layers sandwiched between conducting layers.
By stacking the TMTTF molecules from neighboring layers along the direction normal
to the layer, electron conduction becomes one–dimensional (1D). Within the donor
layer, by contrast, the BEDT − TTF molecules are stacked in two dimensions (2D).
The insulating anion layers separate donors from each other in the interlayer direction.
Due to this stacking procedure, 2D conduction can occur in the layer, which results
in a smaller transfer integral in the interlayer direction than the in–plane direction.
Organic conductors κ–(BEDT − TTF)2 X are quasi–two–dimensional structures in
which the repeat unit is composed of an insulating anions layer X and a layer of conducting BEDT − TTF molecules (see Figure 2.1a). The BEDT − TTF molecules are
labeled with Greek letters to indicate their packing arrangement. The face–to–face
molecular dimers in κ–type salts are oriented roughly at 90 degrees relative to their
neighbors (see Figure 2.1b). These salts contain different anions X that can penetrate
into the insulating layer and change the distance between BEDT − TTF dimers [34].
19
a
b
Figure 2.1: (a) The κ–(BEDT − TTF)2 X crystal structure. (b) The κ–type salt packaging arrangement. Taken from Ref. [34].
Particularly, X = NH4 Hg(SCN)4 and Cu(NCS)2 salts have been extensively investigated. The crystal structure of κ–(ET)2 Cu(NCS)2 at room and low temperature (104
K) has been reported for the first time by the Uramaya’s group [53] and is shown
in Figure 2.2a. In this lattice, the space group is P 21 and the center of inversion is
not present. Table 2.1 provides the crystallographic data at room temperature and
Tc for the organic superconductors κ–Cu(NCS)2 and α–NH4 [48]. It is the packing of
donor molecules within the highly conducting layer (here the bc plane) in the κ–phase
that distinguishes it most from the other crystal structures. Dimers are formed when
neighboring pairs of the ET molecules are rotated about 90 degrees relative to each
other and approximately 45 degrees relative to their b and c axes (see Figure 2.2b) [48].
20
a
b
c
Figure 2.2: The κ–(ET)2 Cu(NCS)2 unit cell. (a) The κ–(ET)2 Cu(NCS)2 crystal structure. The
stacking the planar organic ET molecules vertically in the bc–plane with the inorganic anions forming
insulation sheets between layers. The carbon atoms are represented by full circles and the sulfur
atoms by open circles. (b) An arrangement of the ET molecules along the a–axis. Stacking of
dimer pairs is visible within each conducting layer along the a–axis of the crystal. This stacking
arrangement causes the conductivity to be almost isotropic within each plane. (c) An illustration of
the Cu(NCS)2 anions arranged along a–axis. Taken from Ref. [22, 48, 53].
A separating layer of insulation is created by the anion, here a V–shaped Cu(NCS)2 – ,
between adjacent ET sheets. A weak connection between the anions and the Cu–S
bonds forms polymer–like zig–zag chains in κ–(ET)2 Cu(NCS)2 . Since oriented anion
molecules have a V shape, there is no center of symmetry resulting in a P21 space
group for κ–(ET)2 Cu(NCS)2 [48].
Figure 2.3a reveals the crystal structure for the α–NH4 salt [54, 50]. This salt
has a layered structure, where a BEDT − TTF donor layer and an anion sheet stack
alternately along the b-axis, which is a characteristic structure for an organic conducTable 2.1: The room temperature crystallographic data and the Tc of organic superconductors
κ–Cu(NCS)2 and α–NH4 . The compounds have the form phase–(ET)2 X [48].
a(Å)
b(Å)
c(Å)
α(°)
β(°)
γ(°)
Space group
V(Å3 )
Z
Tc (K)
κ–Cu(NCS)2 [53]
16.256
8.456
13.143
90
110.28
90
P21
1694.8
2
10.4
α–NH4 Hg(SCN)4 [54]
10.091
20.595
9.963
103.67
90.47
93.30
P1
2008
2
1.1
Phase–X
21
tor. The crystal comprises three kinds of BEDT − TTF molecules (one BEDT − TTF
molecule (A), two halves of BEDT − TTF molecules (B and C)), one NH4 cation, and
one [Hg(SCN)4 ] anion. Its triclinic structure has three unit cell angles α = 103.67°,
β = 90.47°, and γ = 93.30° (see Table 2.1). A 2D network of the side–by–side S–S
contacts of the donor molecules is formed in the ac–plane. Figure 2.3b shows the arrangement of the donor molecules along the b–axis. The ”fish–bone” arrangement of
the ET molecules in the conducting layer as well as the thick three–layered polymeric
anion, two thiocyanate layers and a layer separated by Hg2 + and NH+ are the two of
the most significant structural characteristics. There are two kinds of stacking modes
along the c–axis (I and II); in the mode I, the ET stacking molecules, A1 and A2 , are
b
a
c
Figure 2.3: (a) The α–(ET)2 NH4 Hg(SCN)4 crystal structure projected along the c–axis at 298 K.
(b) The α–(ET)2 NH4 Hg(SCN)4 molecular arrangement projected along the b–axis. Here, the ET
donors are just displayed. (c) The anion molecular arrangement for The α–(ET)2 NH4 Hg(SCN)4 .
Taken From Ref. [54, 50].
22
related to each other by an equivalent inversion center, whereas in the mode II, the
ET donors B and C are on the inversion center and stack alternately (nonequivalent
positions). As a result, there are two formula units in the unit cell. The Hg systems
have a curious anion arrangement. The anion layer comprises a triple–sheet (y = 0,
±0.08) parallel to the ac–plane. The y = 0 sheet containing Hg2 + and NH4 + cations
is sandwiched by y = ± 0.08 sheets where linear SCN groups are located. As shown
in Figure 2.3c, the Hg atom is tetrahedrally coordinated by two SCN atoms of the
SCN groups on the upper y = +0.08 sheet and another two SCN atoms on the lower
y = −0.08 sheet. The two coordinated SCN groups on the upper (or lower) sheet
are parallel to each other. Also, the two upper–coordinated SCN groups are arranged
almost perpendicularly to the other two lower–coordinated SCN groups [50].
2.3
Band Structure
Despite their complex molecular arrangements, organic conductors possess simple
Fermi surfaces. The Fermi surface of most salts is either 1D or 2D. Organic metals
such as TMTSF have intermolecular electronic overlap caused by π electrons resulting
in a quasi–1D band in the stacking direction. Based on the structures illustrated in
section 2.2, it is evident that there is a drastic difference between phases in the overlap
for the ET salts. The alignment of the ET molecules side–by–side or face–to–face
influences the overlap integral, which affects the transfer energies considerably. As a
function of angle, the overlap integrals of HOMO bands of two neighboring molecules
of TTF or ET have been measured at any fixed distance by T. Mori [55].
The transfer energy can be calculated from the overlap integral. With the appropriate values of the energy, knowledge of the crystal structure, and using the usual
tight–binding approximation, the band structures and the Fermi surfaces of organic
23
superconductors κ–(ET)2 Cu(NCS)2 and α–(ET)2 NH4 Hg(SCN)4 have been obtained.
In the organic superconductors κ–(ET)2 Cu(NCS)2 and α–(ET)2 NH4 Hg(SCN)4 , knowledge of the band structures, the crystal structure, the Fermi surfaces, and the tight–
binding approximation have been derived from the proper energy values. A semi–
empirical approach is usually used to calculate the bands for these salts, which is
tight–binding extended Huckel method [39, 40, 41]. The Fermi surfaces of organic conductors can be identified qualitatively using this method. Some parameters, however,
like the bandwidth and the effective band masses, pose serious problems. Researchers
have recently used first–principles calculations to study the electronic structures of
molecular metals. Charge transfer salts BEDT − TTF are formed by the transfer of
0.5 electrons from one BEDT − TTF molecule to an anion X. Due to dimerization,
κ–(BEDT – TTF)2 X salts are considered half–filled systems. The monovalent anions
X receive an electron from the two molecules. Figure 2.4 illustrates the hybrid orbitals
created by dimerization that include bonding and anti–bonding orbitals [34]. A transfer integral is represented here by the tdimer within the dimer. During the occupancy
of the dimer by three electrons, a hole is created which increases the conductivity of
the dimer.
Figure 2.4: Illustration of orbital hybridization in a schematic form. Taken from Ref. [34].
24
The electrons at the Fermi surface (FS) determine the electronic properties of superconductor κ–(ET)2 Cu(NCS)2 , as is the case for ordinary metals. The energy band
structure has been calculated first by Oshima et al. [35] employing a simplified tight–
binding approximation. The calculation of the electronic structure is greatly simplified due to the much stronger intermolecular interactions than those between adjacent
molecules. The valence orbitals of the atoms of a single BEDT − TTF molecule are
generalized into an orthogonalized set of wave functions through a method referred
to as the extended Huckel approximation [56]. This basis set is predominantly constructed out of the orbitals of the four sulfur atoms near the central C–C bond. The
highest occupied molecular orbital (HOMO)(only the electrons(holes) near the FS
are taken into account) from each BEDT − TTF molecule is then used in the tight–
binding approximation. The unit cell of this material contains four BEDT − TTF
molecules and is therefore a four–point basis with four bands near the Fermi level.
Each BEDT − TTF HOMO has two electrons and donates 1/2 electron to the anion
layer and this leads to a three–quarter filled conduction band. The conduction bands
are split up by the strong dimerization of the ET molecule, resulting in a half–filled
upper band. Figure 2.5a shows the Fermi surface of the κ–(ET)2 Cu(NCS)2 as it results from tight–binding calculations [35]. In these calculations, interlayer electron
transfer is not taken into account, therefore the FS is strictly two–dimensional. In
the unit cell, there are four BEDT − TTF molecules represented by the four bands.
There is an energy gap at the boundary of the Z–M zone due to the lack of an inversion center in the κ–(ET)2 Cu(NCS)2 crystal lattice. As a result of the gap openings,
the FS contains two open quasi–1D corrugated sheets and closed hole–like quasi–2D
orbits (α pockets), producing an in–plane anisotropy of this q2D material. The con1
:1:1.2.
duction anisotropy has been determined by Saito [57] to be σa : σb : σc =
600
It has been found that at high magnetic fields [58] the electrons can tunnel the gap
25
and form larger closed orbits called β orbits.
Figure 2.5b illustrates a calculation of the band structure and the Fermi surface for
the α–(ET)2 NH4 Hg(SCN)4 salt based on the crystal structure at room temperature
(298 K) [50, 59]. In a similar way to the κ–phase, 1D and 2D bands coexist as
two lightly warped open sheets with the surface vector running roughly aligned with
the a direction and a closed hole–like orbit in the corner of the Brillouin zone. As
opposed to the κ–phase, the 1D open sheets in the α–(ET)2 NH4 Hg(SCN)4 salt have
almost perfect nesting. The calculated HOMO energy levels of crystallographically
independent three donor molecules show no meaningful difference, indicating the
absence of any charge separation among the donor molecules. The calculated overlap
a
b
Figure 2.5: (a) A calculation of the band structure and the Fermi surface for the superconductor
κ–(ET)2 Cu(NCS)2 . Taken from Ref. [35]. Γ corresponds to the center position of the first Brillouin
zone (0,0), Y = (π/b, 0), Z = (0, π/b) and M = (π/b, π/c). (b) A calculation of the energy dispersion
and the Fermi surface for the superconductor α–(ET)2 NH4 Hg(SCN)4 . Taken from Ref. [50]. The
overlap integrals are c1 = −1.9, c2 = 6.8, c3 = −1.1, c4 = −1.4, p1 = −10, p2 = −9.7, p3 = 13.3,
and p4 = 13.2 × 10−3 .
26
integrals of donor HOMO are c1 = −1.9, c2 = 6.8, c3 = −1.1, c4 = −1.4, p1 = −10,
p2 = −9.7, p3 = 13.3, and p4 = 13.2 × 10−3 . The transverse interactions (p1–p4) are
larger than those along the stack (c1–c4). Therefore, the two-dimensional network of
these predominant transverse interactions (p1–p4) affords a two-dimensional energy
band. The tight–binding energy band structure calculated from these overlap integrals
is provided in Figure 2.5b. The two-dimensional Brillouin zone has a pair of electronlike open 1D Fermi surfaces that run along the c–direction, and a closed 2D hole–like
Fermi surface around the V point.
27
Chapter 3
Theory of Magnetic Quantum Oscillations
The first observation of magnetic quantum oscillations (MQOs) owing to quantization
of the electron Landau level was published in 1930 by de Haas and van Alphen
(dHvA) in the magnetic susceptibility [60] and by Shubnikov and de Haas (SdH)
in the resistivity [61] of Bismuth. Previously, Landau had predicted the de Haas–
van Alphen effect in a paper on the diamagnetism of metals [62] but was skeptical
about its experimental detection due to sample inhomogeneities. The theoretical
descriptions of the FS investigation technique in simple metals were first reported
in the 1930s by Peierls [63], Shoenberg [64, 65] and Landau [62]. The quantization
of the electron orbits into Landau levels, along with their magnetic field–dependent
position with respect to the Fermi level, gives rise to an oscillation of the electron
density of states. Hence, as a Landau level crosses the Fermi level with changing
magnetic field, any quantity which depends on the density of states at the Fermi level
will be affected. This results in the oscillations of magnetoresistance, magnetization,
and other quantities. In order to further understand the dHvA effect, it was necessary
to examine the effect of temperature and impurities. After the generalization of the
Shoenberg and Landau theory by Lifshitz and Kosevich (LK) for an arbitrary electron
28
spectrum and Fermi surface, considerable progress was made.
The measurements of dHvA and SdH effects in understanding metals’ electronic
properties have greatly improved due to the development of superconducting magnets
with large and homogeneous fields as well as the availability of pure samples. An
important question that arose after the discovery of superconducting materials was
how their electronic structure looked in the different compounds. The dHvA and SdH
effects cannot be used to explore the FS because of the lack of closed electron orbits
in 1D materials [48]. Interestingly, it has been shown that the topological aspects of
the 1D FS can be obtained by measuring the angular dependence of the resistance
in high magnetic fields and at very low temperatures [48]. In contrast, 2D organic
metals are best suited for studying dHvA and SdH effects. A steady increase has been
observed in the number of investigations into magnetic quantum oscillations since the
first discovery of SdH oscillations [35, 36, 37] in ET compounds [48].
In this chapter, the theoretical framework necessary to work from the quantum
oscillations to a description of the Fermi Surface as well as to the understanding
of different phenomena in the normal state is presented. Using a semiclassical and
a quantum approach, we show what determines the magnetic quantum oscillations
observed in quantities such as magnetoresistance, magnetization, and heat capacity.
We also summarize the basic derivation of the Lifshitz–Koshevich–Shoenberg theory.
Based on this theory, in Chapter 4, we analyze the information contained in the
quantum oscillations for the organic superconductors κ–Cu(NCS)2 and α–NH4 .
29
3.1
3.1.1
The Origin of Quantum Oscillations
Landau Quantization
In a magnetic field, electron orbits are quantized by Bohr–Sommerfeld’s relation
I
p⃗. d⃗r = (n + γ)2πℏ,
(3.1)
where the canonical conjugate momentum is represented by p⃗ and ⃗r are position
variables. Then,
I
I
p⃗. d⃗r =
q
ℏ⃗k. d⃗r +
c
I
⃗ d⃗r.
A.
(3.2)
The Lorentz force is well–known for describing electron motion with charge q in a
magnetic field as follows:
ℏ
q d⃗r
d⃗k
⃗
=
× B.
dt
c dt
(3.3)
The following result can be obtained by integrating with respect to time
q
⃗
ℏ⃗k = ⃗r × B.
c
(3.4)
Taking Equation 3.4 into account, Equation 3.2 can be rewritten by expressing the
magnetic flux Φ in the following way:
I
I
p⃗. d⃗r =
q
ℏ⃗k. d⃗r +
c
I
⃗ d⃗r = − 2q Φ + q Φ = − q Φ.
A.
c
c
c
(3.5)
Combining Equations 3.1 and 3.5, we can deduce the quantized magnetic flux as
follows:
Φn = (n + γ)
30
2πℏc
,
e
(3.6)
where the magnetic flux quantum is represented by
3.1.2
2πℏc
.
e
Quantum Oscillation Frequency
Based on Equation 3.3, the relationship between the areas Sn in k–space and An of
the orbit in real space can be expressed as follows:
An = (
ℏc 2
) Sn .
eB
(3.7)
The relation between Sn and B can be obtained by noting that Φn = BAn ,
Sn = (n + γ)
2πe
B.
ℏc
(3.8)
The Equation 3.8 can be written for two consecutive orbits, n and n + 1 as
Sn
2πe
1
= (n + γ)
Bn
ℏc
(3.9)
and
Sn+1
1
Bn+1
= (n + 1 + γ)
2πe
.
ℏc
(3.10)
In k–space, the area of the electron Fermi surface (S) can be written as follows,
S(
1
Bn+1
−
1
2πe
)=
.
Bn
ℏc
(3.11)
where Sn and Sn+1 are the same. The oscillations of electron density at the Fermi
1
surface is determined by .
B
Electrons are affected by the Lorentz force when they are in a magnetic field.
Consequently, electrons move in closed orbits, where their quantized energy spectrum
31
a
b
Figure 3.1: An illustration of the Landau tubes for (a) a 3D isotropic cylindrical Fermi surface
and (b) a quasi–two–dimensional (q2D) warped cylindrical Fermi surface. Taken from Ref. [51].
is represented by the Bohr–Sommerfeld quantization. L. D. Landau [62] was the first
to propose these orbits and to determine the spectrum of a free electron gas in a
magnetic field
ℏ2 k∥2
1
,
E(n, k∥ ) = (n + )ℏωc +
2
2me
(3.12)
where n = 0, 1, 2, ... , the k∥ is the wave vector component parallel to the field B,
eB
the ωc =
is the cyclotron frequency of a free electron, and me is the free electron
me
mass. As a result, the allowed states in k–space are confined to coaxial tubes, known
as Landau tubes, parallel to the magnetic field direction, resulting in degeneracy
of the states on each tube. As the field increases, these Landau tubes increase in
cross–section area in accordance with the Onsager relation [66],
1 2πeB
.
Sn = (n + )
2 ℏ
(3.13)
Figure 3.1 shows Landau tubes for a three–dimensional (3D) isotropic cylindrical FS
and a quasi–two–dimensional (q2D) warped cylindrical FS. A motion of the Landau
tubes is observed crossing the FS with a constant period in B −1 –scale as the field
32
changes
1
2πe
1
∆( ) =
= ,
B
ℏSi,k
Fi
(3.14)
where Si,k is the extremal cross section area. As a result, the density of states near
the FS oscillates. The orbitals on the FS enclosing a locally–extremal momentum
space area Si,k determine quantum oscillations with fundamental frequencies Fi , where
ℏ
Si,k .
Fi =
2πe
Several physical properties, such as magnetization, conductivity, heat capacity,
and thermoelectric power, also start oscillating as a result of state density near the
FS. Magnetic quantum oscillations are most often studied using two properties: magnetization, which was firstly discovered by W.J. de Haas and P.M. van Alphen [60],
and electrical resistance, which was firstly observed by L.W. Shubnikov and W.J. de
Haas [61]. Due to this, de Haas–van Alphen (dHvA) oscillations and Shubnikov–de
Hass (SdH) oscillations are considered for the oscillations of magnetization and resistance, respectively. Ref. [64] provides a more detailed explanation of the magnetic
quantum oscillations. These relevant aspects of the magnetic quantum oscillations
for the organic superconductors are also summarized in Ref. [51].
3.2
Lifshitz–Kosevich (LK) Equation
In section 3.1, we have seen how the electron density of states at the Fermi level
oscillates as a result of the quantization of the electron energy spectrum in a magnetic
field. As a consequence, various thermodynamic and transport properties oscillate,
such as magnetization, heat capacity, conductivity, and elastic constants. These
oscillations are characterized by magnetic quantum oscillations (MQOs). Due to
their ability to provide information about the Fermi surface, electron spin splitting,
33
cyclotron mass, and relaxation time, MQOs are an important tool in analyzing the
electronic properties of metals.
The magnetic quantum oscillations in 3D metals are theoretically explained by
Lifshitz and Kosevich [67]. The principles of derivations are presented in many textbooks on metal theory [65, 68]. Here we will only present the main results of the
theory and its version for describing the De Haas–van Alphen (dHvA) and Shubnikov
de Haas (SdH) effect for quasi 2D materials.
3.2.1
De Haas–van Alphen (dHvA) Oscillations
Schoenberg [65] contributed considerably to the refinement of the de Haas–van Alphen
effect into a powerful probe of the Fermi surface. In order to measure MQOs, two
major techniques have been widely employed. The first, using the principle that as a
magnetized sample moves in a magnetic field, it experiences a torque proportional to
its magnetic moment, as a result of which a change in magnetization can be measured
by measuring the oscillations in the angular position of the metallic sample. Pulsed
magnetic fields induce a voltage in a pickup coil surrounding the sample in the second
technique. Due to the proportionality of induced voltage d/dt = (dM/dB)(dB/dt),
the oscillations in susceptibility χ = dM/dB can be measured as functions of field.
The MQOs have been attracting a great deal of attention based on the theoretical
successes attained in formulating them. The 3D Lifshitz and Kosevich (LK) model [67]
was among the most important contributions. A brief discussion of the important
equations for the 3D LK model as well as, an extension of the scope to the 2D SdH
MQOs are presented in this section.
In k–space, electronic semi–classical orbits are constrained to planes perpendicular
to the magnetic field B̂ under the influence of a strong external magnetic field and
34
surround k–space area Sk . Landau quantization further confines them to laying on
Landau tubes, that expands with an increase in a magnetic field. The MQOs can be
observed in magnetic torque and magnetoresistivity due to the diverging density of
states as Landau tubes cross the Fermi surface. The Sk varies with k∥ in 3D metals,
where k∥ is the k component parallel to the magnetic field. The frequency and phase
of oscillations will therefore vary for each slice at a given k∥ . It is the sum of these
contributions that determines the total signal. The Fermi surfaces with extremal
cross–sections where dSk,i /dk∥ = 0 usually dominate the sum. In this way, the MQOs
have a fundamental frequency Fi given by orbits on the FS that are surrounded by
ℏ
Si,k [65]. It is generally
locally–extremal momentum space areas Sk,i , where Fi =
2πe
the case that complex Fermi surfaces are characterized by many extremal orbits that
contribute to the total oscillatory response.
Rather than the number of electrons N , the chemical potential µ is used to calculate magnetization from thermodynamic quantities. The grand thermodynamic
potential was therefore used by Lifshitz and Kosevich [67] as
Ω = G − N µ,
(3.15)
where G represents the free energy. Using the ideal case of no additional smearing
effects, the oscillatory part of the thermodynamic potential was calculated as
Ωosc =
∞
X ∂ 2 Si,k
X
e5/2 V B 5/2
Fi π
π
−1/2
−5/2
|
|
p
R
R
R
.cos(2πp(
−
)
±
),
T
D
S
2
23/2 me ℏ1/2 π 7/2 orbitsi ∂k∥
B
2
4
p=1
(3.16)
where B is the field, Si,k shows the Fermi surface extremal area, V refers to a sample’s volume in real space, p represents the MQOs’s harmonic number, and Fi is the
MQOs’s frequency. In the first sum, all the extremal Fermi surface areas Sk,i perpen-
35
dicular to the applied field are considered, whereas, in the second sum, the harmonics
p of each fundamental oscillation frequency are encompassed [69]. Various damping
π
terms are RT , RD , and RS , while the ± represents a correction arising from hole
4
carriers on orbits with a maximum or minimum cross–section in 3D samples [69].
In the case of constant temperature T and constant chemical potential µ, the
magnetization parallel to the field is calculated as the field derivative of the Gibbs
thermodynamic potential µ,
−
→
∂Ω
M = −( →
− )T,µ .
∂B
(3.17)
dΩosc
)T,µ , can be
dB
calculated using the sum of the harmonics assuming the FS has only one extremal
The oscillatory part of the magnetization along B, M∥,osc = − (
cross–section perpendicular to B as
M∥,osc
∞
X
e5/2 V B 5/2
∂ 2 Si,k −1/2 X −3/2
Fi 1 π
= − 1/2
Fi |
|
p
RT RD RS .sin(2πp( − )± ).
2
1/2
5/2
2 me ℏ π orbitsi
∂k∥
B 2 4
p=1
(3.18)
The perpendicular magnetic moments are calculated from oscillations as follows
M⊥,osc,i = −
3.2.2
1 dFi
M∥,osc,i .
Fi dθ
(3.19)
Shubnikov–de Hass (SdH) Oscillations
Unlike dHvA oscillations, the Shubnikov–de Haas (SdH) oscillations are more complicated and not fully understood, since a possible modification of scattering processes
caused by a quantizing magnetic field must be explicitly considered [70]. For most
applications, however, it is sufficient to consider Pippard’s principle [49] that densities
of states around the Fermi level are proportional to scattering probabilities. The 2D
dHvA equations can be applied to 2D SdH cases by stating that the oscillatory part
36
of magnetoresistance is ρosc ∝ Nosc ∝ B 2 (dM/dB), where ρ represents the resistivity,
Nosc is the density of states, B shows the magnetic field, and M is the magnetization.
Consequently, the oscillations of magnetoresistance can be described as follows
∞
2
X
X
Fi 1
ρosc
π
1/2
2 ∂ Si,k −1/2
p−1/2 RT RD RS .cos(2πp( − ) ± ). (3.20)
|
∝ B me
Fi |
2
ρ0
∂k∥
B
2
4
p=1
orbitsi
The SdH oscillations are influenced by reduction factors in the same way as the dHvA
oscillations because of the direct proportionality. Below we introduce the damping
factors RT , RD , and RS to analyze quantitatively the SdH oscillations in 3D metals
using Equation 3.20.
3.2.3
Damping Terms
The effects of the finite temperature T , finite relaxation time τ , and electron spin
are introduced as a ”phase smearing” [65, 22, 68, 71]. One can imagine the effect
of all these features as a superposition of oscillations like in Equation 3.20 in which
the frequency Fi is varied within a small range around the idealized value. In the
following, we will describe each of these effects in some detail.
The Finite Temperature Effect–The damping factor RT (p) appears as a result
of temperature–induced smearing of the Fermi distribution function [65]. Non–zero
temperature implies a spread in electrons around the Fermi level as opposed to a
single well–defined energy. Intuitively, the Fermi energy ϵ0f of metal at temperature
T can be considered to be equivalent with a distribution of hypothetical metals at
T = 0 having Fermi energies µ spread around ϵ0f . Each of these metals would have
an oscillation frequency Fi ∝ Si,k slightly different from each other and this should
result in smearing of the phase of the oscillations and a reduction in the amplitude
of the oscillations. Lifshitz and Kosevich have calculated the exact expression of the
37
thermal damping RT (p) is given by
RT =
X
sinh(X)
with
X=
2π 2 kB T pm∗
,
eℏB
(3.21)
Plugging all known constants in Equation 3.21 we can rewrite it as
RT =
14.7pm∗ T /B
,
sinh(14.7pm∗ T /B)
(3.22)
where kB is Boltzman’s constant and m∗ (m∗ = mc /m0 ) is the quasiparticle effective
mass that can be renormalized using electron–electron and electron–phonon interactions. The RT (p) is dependent on the ratio X ∝ kB T /ωc with the cyclotron frequency
ωc = eB/m∗ , so in this case, a large damping will occur when the thermal broadening
is higher than or equal to the distance between adjacent Landau levels. In order to
observe oscillations, both low temperatures and high fields must be present to reduce the thermal damping. Equation 3.22 specifies the temperature dependence of
the oscillation amplitudes of magnetoresistance (Equation 3.20) and magnetization
(Equation 3.18) that is illustrated in Figures 3.2a and 3.2b for different simulation
parameters. When the temperature is increased, a decrease in the amplitude of the
magnetoresistance is observed [72]. The temperature dependence of the magnetoresistance amplitude allows one to determine the cyclotron mass of the studied material
by fitting the Equation 3.22 to the measured amplitude of the magnetoresistance
oscillations at a fixed magnetic field.
The Finite Relaxation Time Effect–The damping factor that takes into account the finite relaxation time RD (p) was first derived by Dingle [73, 74]. The
Dingle term refers to the scattering of electrons in the presence of impurities. In
this study, he demonstrated that the Landau levels no longer appear sharp when
38
a
b
Figure 3.2: Computed RT (p) as a function of magnetic field and temperature based on Equation 3.22. (a) The value of p and m∗ in this simulation are 1 and respectively 3.5. For κ–Cu(NCS)2
this would correspond to the α orbit. (b) The value of p and m∗ in this simulation are 1 and respectively 6.99. For κ–Cu(NCS)2 this would correspond to the magnetic breakdown orbit, called the β
orbit. Taken from Ref. [72].
electron scattering is considered (unlike in the perfect crystal where no scattering is
supposed), but rather broadened based on the uncertainty principle. As a result, the
amplitude decreases, similar to what happens when the temperature rises. Dingle
used the Lorentz distribution function to describe the broadening and he obtained
the following expression for the
πpm∗
2π 2 pm∗ kB TD
14.7pm∗ TD
RD = exp(−
) = exp(−
) = exp(−
),
eBτ
eℏB
B
39
(3.23)
where τ represents the scattering time, m∗ is the band (cyclotron) mass, and TD =
ℏ
indicates the Dingle temperature. In essence, a finite scattering time leads to
2πkB τ
an expansion of sharp quantum levels into Lorentzian regions of finite width. Smearing of the energy levels produces a suppression of oscillation amplitude, just as does
the finite temperature. Figures 3.3a and 3.3b show the theoretical behavior of Dingle
factor as a function of magnetic field. A Dingle temperature can be determined by
computing effective mass from the temperature dependence of SdH amplitude and
the field dependence of the amplitude. The crystal purity can be determined by the
Dingle temperature. A low Dingle temperature value indicates a clean sample while
a
b
Figure 3.3: The finite relaxation time effect. RD (p) as a function of magnetic field and Dingle
temperature for p = 1 and (a) m∗ = 3.5, and (b) m∗ = 6.99. Taken from Ref. [72].
40
a
b
Figure 3.4: Schematic illustrations of (a) the Zeeman splitting of the Landau levels and (b) the
split of the MQO’s amplitude in a high magnetic field. Taken from Ref. [75].
a high value indicates a dirty sample. The procedure for TD calculation is exposed in
Chapter 4.
The Electron Spin Effect–The electron spin moment interacts with the magnetic field and produces a Zeeman splitting of each Landau level into two (see Figure 3.4). The energy difference between the formerly spin–degenerate levels is
41
∆E = gµB B,
(3.24)
where g represents the splitting factor (g–factor) and µB indicates the Bohr magneton.
For free electrons g = 2.0023, but this value can be different because of the effect of
spin–orbit coupling on the spin moment. The two sets of Landau tubes will pass the
Fermi surface at different fields with a phase defined as
∆ϕ =
2π∆E
,
ℏωc
(3.25)
and each will contribute to the magnetoresistance with waveforms (spin–up, spin–
down) having the same fundamental frequency but half the total amplitude in the
absence of the spin interaction. With Equations 3.24 and 3.25, we can calculate the
relative displacement of the two waveforms in inverse field space for the extremal
section as
g ∗ 2πe
g ∗
m
=
m
2 ℏcS
2F
(3.26)
Therefore, we expect the magnetoresistance to become
g ∗
ρosc 1
g ∗
1 ρosc 1
[
( +
m )+
( −
m )]
2 ρ B 4F
ρ B 4F
(3.27)
With simple trigonometric manipulation, it is readily seen that the net result is a
modification of Equation 3.20 which is obtained by multiplying each term in the
summation by the spin–splitting factor RS (p),
RS (p) = cos(
pπgm∗
)
2
(3.28)
where renormalization of the g–factor and effective cyclotron mass is achieved through
42
electron–electron and electron–phonon interactions. In practice, if m∗ = mc /m0 is
known from the temperature analysis of the magnetoresistance, from Equation 3.28
one can evaluate the g–factor. Moreover, comparing its value with the value gs
obtained in electron–spin–resonance experiments one can extract information about
the influence of many–body interactions on spin splitting. In tilted magnetic fields
the effective mass increases proportional to 1/cosθ. Taking into account this θ–
dependence in Equation 3.28, we find that if the phase difference pgm∗ /cosθ for p = 1
is an odd multiple of π, RS (p) becomes zero and thus we can precisely determine the
product gm∗ .
43
Chapter 4
Data Analysis and Results
As mentioned in Chapter 3, when an electron is in a magnetic field, its energy spectrum is quantized. Increasing the magnetic field causes the quantized energy levels
to periodically intersect the Fermi level, resulting in an oscillation of the electron
density. Consequently, various thermodynamic and transport properties oscillate, including magnetization (de Haas–van Alphen effect), resistivity (Shubnikov–de Haas
effect), heat capacity, elastic constants, etc. In quantum mechanics, these oscillations
are referred to as quantum fluctuations. By using the TDO technique, the in–plane
resistivity of electrons can be measured with the in–plane RF penetration depth measurement. The Shubnikov–de Haas Quantum Oscillation (SdH QO in short) can thus
be measured in a normal state.
When analyzing magnetoresistance oscillations (SdH QOs), in our quest to understand transport and superconducting phenomena and to determine the normal
state properties of organic superconductors, it is key to know the topology of the
Fermi surface and the effects that factors such as temperature, magnetic field, and
crystal orientation have on these oscillations. In this chapter, the measurement of the
SdH QOs in the organic superconductors κ–Cu(NCS)2 and α–NH4 will be discussed.
44
First, we show that our Fermi surface measurements are in agreement with the existing theoretical and experimental results. The effects of various factors affecting
the magnetoresistance oscillations are also shown. Then, the angular dependence of
the frequencies and of the amplitudes of the magnetoresistance oscillations, which
agrees with the theoretical predictions, is presented. Finally, a novel method, the
Filter–Peaks–Fit (FPF) method, is presented to analyze the experimental data and
to calculate the damping factors such as the effective mass m∗ , the g–factor, and
Dingle temperature TD for different orbits.
4.1
Damping Factors Calculation
As the magnetic field increases, new frequencies associated with the different orbits
appear in the SdH data due to the magnetic breakdown effect. The total amplitude
will be the sum of the contributions given by all possible frequencies α, 2α, β, β − α,
β + α, β + 2α and their harmonics. In previous studies, different methods have been
used to extract the normal state properties such as the Dingle temperature TD , the
effective mass m∗ , and the spin splitting factor (g–factor). A common approach [76,
77] is to find the amplitudes of each frequency as a function of the field by doing
a series of FPFs on intervals of the data in 1/B space. The Dingle temperature is
found by fitting the amplitude of the α data at low fields to the LK formula, essentially
assuming B0 = ∞. In other studies [58], a full fit was attempted of the raw data
to a sum of terms in the LK formula. Harrison [78] used a simultaneous fit to both
the first and second harmonic of α. In the magnetic breakdown limit, the probability
that the electrons travel on the 2α orbit is lower compared to the probability for the
β orbit, causing the amplitude of the 2α–frequency to be lower than the amplitude
of the β–frequency (see Figure 4.3b) and more susceptible to noise. Therefore, in our
45
study, we primarily analyze the amplitude of the different orbits data. We also show
that the analysis of the different orbits data does not give a unique value for damping
factors.
There are three major problems when analyzing experimental data using the LK
theory and the conventional analysis methods, the first one is the measured SdH
oscillations contain more than one fundamental frequency and harmonics of those
frequencies (plus noise), and it is, therefore, hard to analyze and in particular hard
to fit all of the information at once. The second problem is the oscillatory component
of the amplitude contains values such as the frequency and the phase which are
not orthogonal and have many local minimums making it difficult for most fitting
routines to fit the LK formula with the experimental data. The last problem is that
the damping factors are not uniquely determined using Equations 3.22, 3.23 and 3.28
unless one of them has been previously measured. As will be shown in Section 4.1.1,
the Filter–Peaks–Fit (FPF) method [72] provides an effective technique that solves
all these issues.
4.1.1
Filter–Peaks–Fit (FPF) Method
The conventional analysis methods have some issues when analyzing experimental
data. A first problem is that measured SdH oscillations contain many fundamental frequencies, harmonics of those frequencies, as well as noise, which makes them
difficult to analyze and especially hard to fit all this information simultaneously. Furthermore, since the oscillatory component of the amplitude contains frequencies and
phases that are non–orthogonal and contain many local minimums, most fitting routines find it difficult to fit the LK formula with the experimental data. An effective
technique which is called Filter-Peaks-Fit (FPF) method has been provided by Iz-
46
FFT of the
raw signal
Representing
frequency
vs. 1/B
Frequency Separation
Interpolating
frequency vs.
1/B into a
single wave
3–band pass
filtering
Obtaining orbital signals
Measuring
amplitude
of frequency
Background
subtraction
Obtaining Raw Signal
Filter–Peaks–Fit (FPF)
Figure 4.1: Illustration of the Filter-Peaks-Fit (FPF) method.
abela Stroe [72] to solves all these limitations. Figure 4.1 shows a graph of the FPF
method. The FPF method consists of three steps:
(a) Obtaining raw signals–we start with the split waves corresponding to the
up or down sweeps in a magnetic field (see Figure 4.2a) and representing the frequency
as a function of the inverse magnetic field. Then, we need to interpolate the frequency
versus inverse magnetic field wave into a single wave with a number of points equal
to a power of 2. A reasonable number of points is 213 = 8192, but better results are
obtained when 215 points are used. The x–scaling of the new wave should be set up
for the region of interest, meaning the interval in 1/B space where the oscillations are
observed. The scaling varies from experiment to experiment. For example, for one
set of experimental data, we used an x–scaling between 0.03999 and 0.127 T−1 which
corresponds to a field of 25 and 7.87 T (see Figure 4.2b). Finally, we need to extract
47
b
3.8
8.7°
T = 33 mK
κ–Cu(NCS)2
3.6
3.4
3.2
3.0
4.00
3.95
3.90
22
23
24
25
4.1
Magnetoresistance (arb.units)
4.0
Magnetoresistance (arb.units)
Magnetoresistance (arb.units)
a
Raw signal (θ = 8.7°)
4
9th degree Polyfit
T = 33 mK
κ–Cu(NCS)2
3.9
3.8
3.7
B (T)
0
10
20
3.6
0.03
30
B (T)
0.06 0.09 0.12
1/B (T−1 )
0.15
Figure 4.2: (a) Magnetoresistance versus magnetic field. The inset shows the magnetoresistance
oscillations at magnetic field above 22 T. (b) Magnetoresistance versus inverse magnetic field waves.
The solid line represents the fit with a 9th degree polynomial function.
the background. For this, we fit the interpolated wave with a polynomial function.
In our data analysis, we found that the best fit is given by a 9th–degree polynomial
function (see Figure 4.2b). We then extract the fitted function from the interpolated
wave. Figure 4.3a depicts a raw magnetoresistance signal as a function of inverse
magnetic field. The new obtained wave is further used in the filtering routine.
(b) Frequency separation– A common approach for obtaining electron orbital
frequencies is to find the amplitudes of each frequency as a function of field by doing
a series of FFTs on intervals of the data in 1/B space. Figure 4.3b shows the FFT of
the raw signal. At 33 mK, four fundamental frequencies are observed (Fα , Fβ , Fβ−α
and Fβ+α ) along with one harmonic (F2α ).
(c) Obtaining orbital signl– In the FPF method, the raw signal is first digitally
filtered. This step allows for an independent analysis of each frequency and removes
most of the noise. We used the 3 band pass–filtering routine from the Igor Filter
Design Laboratory by WaveMetrics. The filters are based on a finite impulse response
48
1015
α
25
T = 33 mK
0
−25
−50
0.03
0.06 0.09 0.12
1/B (T−1 )
10
κ–Cu(NCS)2
1013
2α
1012
1011
1010
109
0.15
θ = 8.7°
T = 33 mK
14
β+α
κ–Cu(NCS)2
FFT amplitude (arb.units)
SdH oscillation (arb.units)
Raw signal (θ = 8.7°)
β
b
50
β−α
a
0
2000
4000
SdH Frequency (T)
6000
Figure 4.3: (a) Raw oscillations in magnetoresistance versus inverse field after the background was
subtracted. (b) The FFT of the left signal. The breakdown frequencies are clearly present and are
associated with the possible closed orbits of the Fermi Surface.
filter design.
To filter out each oscillation present in the raw signal we further need to design
the 3 band pass filter by specifying the end of the first band, the start and the end of
the second band, the start of the third band, the second band weight and the number
of terms. Let us suppose that the oscillation of interest has a frequency of Fi . The
first band eliminates all frequencies smaller than Fi and the third band eliminates all
frequencies greater than Fi . The frequency content of the raw signal is determined
through a fast Fourier analysis. After setting up all parameters to obtain the best
frequency response of the filtered frequency, the time response feature of the Igor
routine gives the filtered oscillation. Figures 4.4a and 4.4b show the oscillations for the
α and β frequency after the first step. Similar results are obtained for combinations
of the α and β oscillations. To check the accuracy of the filtered data, the filtered
oscillations are summed to form a new composite signal. The very good overlap
between the composite signal and the original data shows that the filtering algorithm
49
a
b
25
Raw signal (θ = 8.7°)
α-orbit signal
β -orbit signal
T = 33 mK
κ–Cu(NCS)2
0
−25
−50
0.03
50
Raw signal (θ = 8.7°)
SdH oscillation (arb.units)
SdH oscillation (arb.units)
50
0.06 0.09 0.12
1/B (T−1 )
25
0
−25
−50
0.03
0.15
T = 33 mK
κ–Cu(NCS)2
0.06 0.09 0.12
1/B (T−1 )
0.15
Figure 4.4: (a) The α oscillations and (b) the β oscillations vs inverse field after filtering.
is not introducing crucial artifacts into the data. In most cases, the only artifacts
introduced are the ones at the ends of the amplitude data (high and low field), which
we truncated.
4.1.2
Effective Mass Calculation
The effective mass and the g–factor are two important band structure parameters.
The departure of the values of these parameters from the values of the effective bare
band mass, and respectively of the spin splitting factor for free electrons, gives information about the strength of the electron–electron and electron–phonon interactions.
The SdH experiments have proved to be a direct tool to determine the enhancement
of the effective mass and the g–factor. We determine the effective masses of the electrons orbiting the α, and the magnetic breakdown orbits β, β − α, β + α. We find
good agreement with previous experimental results. Based on the obtained effective
masses, we calculate the g–factor for different orbits and find that its value is smaller
compared to the free electron g–factor. This result agrees with the values previously
50
found and with the theoretical predictions.
Band structure calculations give information about the bare band mass. The
effective mass determined from the SdH and dHvA experiments is the bare band mass
renormalized by the electron–electron and electron–phonon interactions. Therefore,
from the value of the effective mass obtained from quantum oscillation experiments,
one can obtain information about the many–body interactions in the material [65]. In
the theoretical framework of the LKS–model, the orbitally averaged effective mass can
be measured from the temperature dependence of the magnetoresistance oscillations
amplitude.
The Effective Mass of the superconductor κ–Cu(NCS)2 –Stroe et al. [72]
used a filtering method, Filter–Peaks–Fit (FPF) method, and determined the effective
masses of the electrons orbiting the α closed orbit as well as the magnetic breakdown
orbits β, β − α, β + α, for the organic superconductor κ–Cu(NCS)2 . Their SdH oscillation measurements performed in fields up to 45 T and temperatures between 0.5
to 4.2 K. The filtering procedure is essential, especially at high temperatures, where
the amplitudes of all oscillations decrease considerably. In the conventional method,
the amplitudes determined from the FFT spectrum are almost indistinguishable from
noise at high temperatures. In the LK formula, Equation 3.22, the finite temperature
Table 4.1: Effective masses determined for the superconductor κ–Cu(NCS)2 [72]. The angle was
0° for Stroe et al. and Meyer et al.’s measurements. Harrison et al. performed the measurements at
13°.
m∗ /m
Orbit
α
2α
β−α
β
β+α
Stroe et al. (0°) [72]
Meyer et al. (0°) [76]
Harrison et al. (13°) [78]
3.4
6.3
3.7
6.99
11
3.2
6.1
–
7.1
10.4
3.5 ± 0.1
6.4 ± 0.6
3.6 ± 0.3
7.1 ± 0.5
8.2 ± 0.4
51
damping factor RT (p) contains the temperature dependence. Equation 3.22 is used
to determine the effective mass by fitting the measured oscillation amplitude to this
dependence. Based on the fit, the effective masses calculated are listed in Table 4.1.
Their results are in good agreement with other experimental results [76, 77, 78]. An
anomaly was obtained for the 2α–effective mass. The measured value is about 7%
lower than the 2m∗α . According to the LK model, the effective mass of the 2α–orbit
oscillations should be twice the effective mass of the α orbit oscillations. This unexpectedly low mass for the second harmonic, also observed in other experiments [78],
was explained as a possible departure of the SdH oscillation from the LK behavior at
high magnetic fields. We will use these results for measuring the Dingle temperature
TD and the g–factor of the superconductor κ–Cu(NCS)2 .
The Effective Mass of the superconductor α–NH4 –We determined the effective masses of the electrons orbiting the α closed orbit as well as the magnetic
breakdown orbits 2α, for a sample α–NH4 . The SdH oscillation measurements were
performed in fields up to 16 T and temperatures between 0.031 to 1.57 K are presented in Figure 4.5. The FFT spectra show that the amplitude of each frequency is
diminished as the temperature increases. In order to accurately extract the amplitude
of each frequency from the total magnetoresistance oscillations, we used the Filter–
Peaks–Fit (FPF) method presented in Section 4.1.2.
Figures 4.6a and 4.6b show
the amplitudes of the α and 2α oscillations after filtering for each temperature. The
m∗ from the SdH QOs can be measured based on the temperature–induced smearing
factor RT defined in Equation 3.22. The log relationship between RT /T and T can
be found from RT as follows
T
RT
≈ −14.7m∗ .
ln
T
B
52
(4.1)
SdH oscillation (arb.units)
θ = 90◦
B
1.57 K
1.25 K
0.74 K
0.50 K
0.15 K
0.09 K
0.031 K
α–(ET)2 NH4 Hg(SCN)4
6
8
10 12 14 16 18 20
B (T)
Figure 4.5: Magnetoresistance oscillations as a function of temperature measured in sample α–
NH4 . The angle between the sample and the magnetic field is θ = 90°.
a
200
·10−3
b
15
·10−3
α–(ET)2 NH4 Hg(SCN)4
α–(ET)2 NH4 Hg(SCN)4
α–orbit oscillations
2α–orbit oscillations
B
10
θ = 90◦
RT (MHz)
RT (MHz)
150
100
0.031 K
50
0.07 0.09 0.11
1/B (T−1 )
θ = 90◦
5
0.031 K
1.57 K
0
0.05
B
0
0.05
0.13
0.74 K
0.07 0.09 0.11
1/B (T−1 )
0.13
Figure 4.6: (a) The α–amplitudes and (b) the 2α–amplitudes at different temperatures for the
effective mass plot.
53
The m∗ is determined by extracting the QOs amplitudes at different magnetic
fields from Figure 4.4, then plotting them as a function of temperature in Figure 4.7.
We determined m∗α to be 2.17 ± 0.13 averaged from 8.31 to 15.79 T based on the
·10−3
a
200
α–(ET)2 NH4 Hg(SCN)4
α–orbit oscillations
θ = 90◦
RT (MHz)
150
15.79 T
100
50
0
8.31 T
0
b
15
0.5
1
T (K)
1.5
·10−3
α–(ET)2 NH4 Hg(SCN)4
2α–orbit oscillations
15.85 T
RT (MHz)
10
5
0
8.10 T
0
0.2
0.4
T (K)
0.6
0.8
Figure 4.7: Amplitude of the QOs for (a) the α–orbit and (b) the 2α–orbit versus Temperature at
different magnetic fields.
54
slopes of the ln(RT /T ) versus T diagram shown in Figure 4.8. In other studies,
similar values, such as 2.2 ± 0.4 at B = 12.1 T, have been reported in Ref. [47]. Also,
the m∗2α is 3.52 ± 0.20 averaged from 8.10 T to 15.85 T. Here there is an anomaly
for the 2α–effective mass.
a
According to the LK model, the effective mass of the
ln(RT /T) (MHz K−1 )
5
B
Avg.
0
m∗
α
= 2.17 ± 0.13
15.79 T
−5
1.76
α–(ET)2 NH4 Hg(SCN)4
−10
b
0
0.5
1
T (K)
8.31 T
1.5
2
ln(RT /T) (MHz K−1 )
0
B
Avg. m∗
2α = 3.52 ± 0.20
15.85 T
−5
1.44
α–(ET)2 NH4 Hg(SCN)4
−10
0
0.2
0.4 0.6
T (K)
8.10 T
0.8
1
Figure 4.8: The ln(RT /T ) versus temperature T diagram which gives (a) m∗α = 2.17±0.13 averaged
from 8.31 to 15.79 T, and (b) m∗2α = 3.52 ± 0.20 averaged from 8.10 to 15.85 T.
2α–orbit oscillations should be twice the effective mass of the α orbit oscillations.
55
But the measured value is about 19% lower than the 2m∗α . Furthermore, the effective
mass parameters increase approximately linearly with the field, reaching 2.41 and
3.84 for α and 2α orbits, respectively, at the highest field window (see Figure 4.9).
In comparison with LK predictions, an increasing effective mass parameter displays
suppressed oscillations amplitude at higher magnetic fields. This unexpected behavior
of the effective mass is explained as a possible departure of the SdH oscillation from
the LKS behavior at high magnetic fields [78].
4.1.3
Calculation of the Dingle Temperature
By considering electron scattering, one can observe a stretching of the Landau levels,
which results in additional smearing of the QOs amplitude very similar to that caused
by finite temperatures. Equation 3.23 defines the smearing factor due to electron
scattering. The temperature TD = ℏ/2πkB τ is known as Dingle temperature where
4.0
α–(ET)2 NH4 Hg(SCN)4
Effective Mass (m∗ )
θ = 90◦
m∗
2α
3.5
3.0
2.5
2.0
m∗
α
12
13
14
15
B (T)
16
Figure 4.9: Effective mass of α and 2α orbits as a function of field.
56
τ displays the electron scattering time.
We investigated the behavior of the Dingle temperature as a function of angle for
the two samples, κ–Cu(NCS)2 and α–NH4 . Previous studies assumed that the Dingle
temperature is the same across all possible orbits. In samples with high scattering,
the assumption of a unique Dingle temperature leads to a contradiction. In our
experiments, we varied the angle from 0 to 50 degrees and found that the Dingle
temperature varies. Also, We find that the Dingle temperature varies from one orbit
to another as has been found in some other systems.
–The Dingle Temperature of the κ–Cu(NCS)2 –To calculate TD , the QOs
amplitude measured from Figure 4.4. These amplitudes are plotted as ln(RD ) versus
1/B. Based on the 2D LK equation of the SdH QOs, the TD is calculated from the
slope in the plot. we measured the Dingle temperature TD for the sample κ–Cu(NCS)2
in different angels and orbits obtained the plots shown in Figure 4.10.
The values
Table 4.2: Dingle temperatures determined for the superconductor κ–Cu(NCS)2 in different angels
and orbits.
Orbit
α
2α
β−α
β
β+α
TD (K)
0°
8.7°
18.7°
40°
Avg. TD (K)
Stroe et al. (0°)17
Harrison et al. (13°)24
0.72
0.48
2.40
2.41
1.27
0.67
0.46
1.70
1.51
0.85
0.82
0.48
1.83
1.50
0.92
0.88
0.71
−
1.81
0.87
0.77 ± 0.09
0.53 ± 0.1
1.98 ± 0.4
1.8 ± 0.4
0.98 ± 0.2
1.7
–
–
1.0
–
0.64 ± 0.04
–
–
–
–
of the Dingle temperatures obtained from the fit are listed in Table 4.2. The Dingle
temperatures TD are estimated for different orbits averaged from 0° to 40° to be 0.77
K ±0.09, 0.53 K ±0.1, 1.98 K ±0.4, 1.8 K ±0.4 and 0.98 K ±0.2, for orbitals α, 2α,
α − β, β, and α + β, respectively. We can observe that the Dingle temperature varies
as the angle and orbit change. Also, there is an anomaly for the Dingle temperatures.
As an example, the Dingle temperature of the 2α–orbit oscillations is not twice the
57
−3
b
ln(RD ) (MHz)
−4
−5
−6
−7
−9
−10
−11
0.03
c
−2
−3
2α
0.06 0.09 0.12
1/B (T−1 )
ln(RD ) (MHz)
0.15
−3
0.82
0.48
1.83
1.50
0.92
−5
−6
−7
−11
0.03
2α
α
β−α
T = 33 mK
θ = 8.7°
0.06 0.09 0.12
1/B (T−1 )
κ–(ET)2 Cu(NCS)2
TD (K)
−5
−6
−7
α
−8
−10
T = 33 mK
θ = 18.7°
0.15
0.15
0.88
0.71
1.81
0.87
−9
2α
0.06 0.09 0.12
1/B (T−1 )
β+α β
−4
−8
−10
α
−8
−2
TD (K)
κ–(ET)2 Cu(NCS)2
β−α
−7
−11
0.03
d
β
−6
−10
T = 33 mK
θ = 0°
β+α
−5
−9
−4
−9
0.67
0.46
1.70
1.51
0.85
−4
α
β+α β β−α
TD (K)
κ–(ET)2 Cu(NCS)2
−3
0.72
0.48
2.40
2.41
1.27
−8
−2
TD (K)
κ–(ET)2 Cu(NCS)2
ln(RD ) (MHz)
−2
ln(RD ) (MHz)
a
−11
0.03
β
2α
T = 33 mK
β+α
θ = 40°
0.06 0.09 0.12
1/B (T−1 )
0.15
Figure 4.10: The ln(RD ) vs. 1/B for different orientations in constant temperature 33 mK. From
the fits, the Dingle temperature TD is calculated for various orbits in (a) θ = 0°, (b) θ = 8.7°, (c)
θ = 18.7°, and (d) θ = 40°.
Dingle temperature of the α–orbit oscillations.
–The Dingle Temperature of the α–NH4 –Similar to the case of sample κ–
Cu(NCS)2 , we measured the Dingle temperature TD for the sample α–NH4 in different
angels for orbits α and 2α obtained the plots shown in Figure 4.11. The Dingle
58
temperature TD is estimated from the slope in Figure 4.11a to be 2.4 T ± 0.3 for the
α–orbit averaged from 0° to 51.2°. For 2α–orbit, the TD is 12.52 T ± 4.3 averaged
from 0° to 41° (see Figure 4.11b). We can observe that the Dingle temperature varies
as the angle changes. Also, We find that the Dingle temperature varies from one orbit
to another as has been found in some other systems. Here there is an anomaly for
the 2α–Dingle temperature. According to the LKS model, the Dingle temperature
of the 2α–orbit oscillations should be twice the Dingle temperature of the α–orbit
oscillations. But the measured value is higher than the 2TD−α .
a
b
0
−4
51.2°
−6
−8
−10
−12
1.71
2.64
2.79
2.16
2.44
2.57
2.53
2.40
T = 33 mK
Avg. TD = 2.4 ± 0.3
α–orbit oscillation
0.07 0.1 0.13
1/B (T−1 )
−5
−6
−7
0.16
−9
TD (K)
9.10
8.20
6.48
5.80
3.88
0°
10.2°
30.7°
41°
20.5°
−8
α–(ET)2 NH4 Hg(SCN)4
0.04
α–(ET)2 NH4 Hg(SCN)4
2α-orbit oscillation
−4
ln(RD ) (MHz)
ln(RD ) (MHz)
−2
0°
−10.2°
10.2°
30.7°
41°
−20.5°
20.5°
−3
TD (K)
T = 33 mK
Avg. TD = 6.69 ± 2.05
0.06
0.07
0.08
1/B (T−1 )
0.09
Figure 4.11: The ln(RD ) vs. 1/B for different angels. From the fits, the Dingle temperature TD
is measured to be (a) 2.4 K ± 0.3 for the α–orbit and (b) 6.69 K ± 2.05 for the 2α–orbit.
59
Chapter 5
Conclusions
In this thesis, we have reported the normal state properties of the organic superconductors α–(ET)2 NH4 Hg(SCN)4 and κ–(ET)2 Cu(NCS)2 . We measured the magnetoresistance quantum oscillations at very high magnetic fields, low temperatures,
and different angles using the contactless TDO measurement technique. Analytical studies on the field, temperature– and angle–dependence of the oscillations show
unusual features such as field– and orbit–dependent effective mass parameter, non–
linear Dingle plots, and orbit–dependent Dingle temperature parameter. Due to the
highly two–dimensional nature of the quantum materials, these results deviating deviates from the Lifshitz–Kosevich formalism. From the measurements, we obtained
the following results for the two organic superconductors:
The Results of the organic superconductor α–(ET)2 NH4 Hg(SCN)4 :
• The effective mass measured value is consistent with the values reported by
other groups.
• There is an anomaly in the effective mass of different orbits. The effective mass
of the 2α–orbit oscillations is lower than the 2m∗α .
60
• The effective mass parameters increase approximately linearly with the field.
• The Dingle temperatures TD varies as the angle changes.
• The Dingle temperature varies from one orbit to another. Here there is an
anomaly for the Dingle temperatures. According to the LKS model, the Dingle
temperature of the 2α–orbit oscillations should be twice the Dingle temperature
of the α–orbit oscillations. But the measured value is higher than the 2TD−α .
The Results of the organic superconductor κ–(ET)2 Cu(NCS)2 :
• The Dingle temperatures TD varies as the angle changes.
• The Dingle temperature varies from one orbit to another. Here there is an
anomaly for the Dingle temperatures. According to the LKS model, as an
example, the Dingle temperature of the 2α–orbit oscillations should be twice
the Dingle temperature of the α–orbit oscillations. But the measured value is
higher than the 2TD−α . There is the same condition for other orbitals.
61
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