UNIVERSITY OF CALIFORNIA Los Angeles
NMR Study on a Quarter-Filled Organic Superconductor – Charge Ordering, Superconductivity, and the FFLO state in a Layered Superconductor
A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Materials Science and Engineering
by
Hsin-Hua Wang
2018 c Copyright by Hsin-Hua Wang 2018 ABSTRACT OF THE DISSERTATION
NMR Study on a Quarter-Filled Organic Superconductor – Charge Ordering, Superconductivity, and the FFLO state in a Layered Superconductor
by
Hsin-Hua Wang Doctor of Philosophy in Materials Science and Engineering University of California, Los Angeles, 2018 Professor Yang Yang, Co-chair Professor Stuart Brown, Co-chair
Due to the low dimensionality of molecular conductors, they are categorized into a class of materials with strong electron correlation, namely Coulomb repulsion. The pairing mech- anism of unconventional superconductors (SCs), which commonly arises in these compounds, is a central and unresolved problem in condensed matter physics. Whereas numerous un- conventional SCs exist in the vicinity of magnetic ground states, we investigate whether magnetic ordering or fluctuation is a necessary ingredient of unconventional superconduc- tivity, as well as whether other mechanisms such as the charge degree of freedom can play a role. Moreover, the low dimensionality of molecular conductors also allows us to study a long-sought-after inhomogeneous high field superconducting phase, the Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO) state. β”-(BEDT-TTF)2SF5CH2CF2SO3 (β”-SC) that belongs to the family of molecular conductors, is an ideal candidate for studies of both topics. Nuclear magnetic resonance (NMR) is one of the few possible microscopic experimental probes for studying molecular conductors. In this dissertation, interesting physics in β”-SC, in the contexts such as charge-fluctuation mediated superconductivity and the fluctuating nature of the FFLO state, is revealed by NMR.
Specifically, our results on β”-SC are consistent with an unconventional superconducting
ii state with singlet pairing and nodal superconducting gap. Moreover, absent or barely distin- guishable evidence of magnetic fluctuations are observed above superconducting transition
temperature Tc, unlike in most other unconventional SCs, including high Tc cuprate SCs. In the normal state, β”-SC is a charge-ordered metal revealed by NMR spin lattice relaxation
−1 rate T1 . The charge disproportionation forms a stripe pattern along the a axis, whereas the
disproportionation increases monotonically upon cooling to Tc. Above Tc, β”-SC exhibits
only moderate antiferromagetic correlation from analysis of Korringa ratio κ ∼ 2. Motion of the ethylene end groups is studied by 2H NMR above 100 K, and is found to induce
−1 13 an enhanced (T1T ) on C through modifying the electron correlation. The response of
−1 (T1T ) to pressure shows that charge disproportionation is suppressed at high pressures as well as Tc according to previous report [1], indicating the possible interplay between charge disproportionation and superconductivity.
In the FFLO state, when the orbital pair breaking effect is suppressed under in-plane mag-
−1 −1 netic field, we observe an unusual enhancement of (T1T ) above the normal state (T1T ) , at the verge of the FFLO–uniform SC phase boundary. Moreover, we observe an enhance-
−1 ◦ ment that is 2.5 times the normal state (T1T ) under 1 misaligned field at B = 9.25 T.
−1 We conclude that both in-plane and out-of-plane (T1T ) enhancements in the FFLO state originate from hyperfine coupling to the electrons. We propose a picture of a fluctuating
−1 FFLO state where the enhancement of (T1T ) is caused by a fluctuating magnetic field induced by dynamics of short-range modulations of the superconducting order parameter.
iii The dissertation of Hsin-Hua Wang is approved.
Dwight Christopher Streit
Yang Yang, Committee Co-chair
Stuart Brown, Committee Co-chair
University of California, Los Angeles
2018
iv To my family, for their unreserved love
v TABLE OF CONTENTS
List of Figures ...... viii
List of Tables ...... xi
Acknowledgments ...... xii
Vita ...... xiii
1 Introduction ...... 1
1.1 Unconventional Superconductivity and Pairing Mechanism...... 1
1.1.1 Magnetically mediated paring? Cases of cuprates and a half-filled system2
1.1.2 Charge ordering and superconductivity arising from electron correla- tion in a quarter-filled system...... 6
1.2 A Novel High Field Superconducting Phase: FFLO State...... 9
1.2.1 Orbital limit and Pauli limit of critical field...... 10
1.2.2 FFLO state and existing experimental evidence...... 12
2 Quarter-filled Quasi-two-dimensional Organic Superconductor β”-(BEDT-
TTF)2SF5CH2CF2SO3 ...... 15
2.1 Crystal Structure...... 15
2.2 Magnetic, Electrical and Optical Properties in Normal State...... 16
2.3 Superconducting Properties...... 18
3 Experimental Technique ...... 21
3.1 Response of Isolated Nuclei to a Magnetic Field...... 21
3.2 How Nuclei Interacts with the Environment...... 23
vi 3.3 NMR Observables...... 25
4 Result I - Charge Ordering and Superconductivity ...... 28
4.1 Applied Magnetic Field Alignment...... 28
4.2 13C NMR Spectra and Spin Lattice Relaxation...... 30
4.2.1 Normal state properties: a charge-ordered metal...... 34
4.2.2 Characteristics of superconducting state...... 39
4.3 Nature of Electronic Correlation: Korringa Ratio...... 42
4.4 Motion of Ethylene Groups...... 47
4.5 Response Under Pressure...... 55
5 Result II: Properties of the FFLO States ...... 58
5.1 Spectroscopic Study of the FFLO state...... 59
−1 5.2 Enhancement of (T1T ) in the FFLO state...... 62
5.2.1 Enhancement under in-plane magnetic field...... 62
5.2.2 Enhancement under out-of-plane Magnetic Field...... 64
−1 5.3 Origin of Enhancement of (T1T ) and the Nature of the FFLO state.... 66
5.3.1 Mechanism of NMR spin lattice relaxation revealed by a molecule-
specific T1 study...... 66
5.3.2 Discussion on the possible origins of relaxation enhancement..... 68
5.3.3 Picture of a fluctuating FFLO state...... 70
6 Conclusion ...... 75
vii LIST OF FIGURES
1.1 Representative phase diagram of Cuprates...... 3
1.2 Effect of electron correlation on a half-filled system...... 5
1.3 Effect of electron correlation on a quarter-filled system...... 6
1.4 Calculated phase diagram of a quarter-filled system on a q2D square lattice based on the extended Hubbard model...... 7
1.5 s and dx2−y2 wave superconducting gap...... 8
1.6 Cartoon of spin- and charge-mediated pairing...... 9
1.7 Illustration of modulated superconducting order parameter...... 11
2.1 Crystal structure and packing motif of (BEDT-TTF)2X compounds...... 17
2.2 Shubnikov–de Haas measurement and Fermi surface calculation of β”-SC.... 18
2.3 Phase diagram of β”-SC extracted from specific heat measurements...... 20
1 3.1 Energy diagram of a spin 2 nucleus under a magnetic field...... 24 3.2 Schematics of NMR spectra of different magnetic ordered phase...... 26
4.1 RF absorption measurement...... 30
−1 4.2 Angular dependence of T1 to align the magnetic field...... 31
4.3 13C NMR spectrum obtained at T = 78 K, B = 8.4530 T for field in the a − b plane 32
4.4 Representative recovery curve of T1 measurement...... 33
−1 4.5 (T1T ) and K plotted as a function of temperature...... 34
4.6 Temperature dependence of intramolecule and intermolecule charge dispropor-
tionation of red and orange molecule extracted from T1 data...... 36
4.7 Susceptibility measurement, K − χ analysis and modeling of NMR shift..... 38
−1 4.8 (T1T ) and K below Tc plotted as a function of temperature...... 40
viii −1 4.9 Comparison of (T1T ) between β”-SC and (TMTSFP)2PF6 ...... 41
4.10 Angular dependent spectra in the a-c and b-c planes and the 2D NMR measure- ment utilized to resolve spectra components...... 42
4.11 Angle dependent shift, dipolar split and Korringa ratio in the a-c and b-c planes 44
4.12 Temperature dependence of the Korringa ratio...... 46
4.13 Eclipsed and staggered conformations of ET molecules...... 47
2 −1 4.14 Ethylene group dynamics studied by H NMR spin lattice relaxation T1 ... 48
2 −1 4.15 Temperature dependence of H T1 at different field...... 50
4.16 Temperature dependence of 13C NMR spin lattice relaxation...... 51
4.17 Temperature dependence of 2H NMR spectra...... 53
4.18 Simulation of 2H NMR spectrum...... 54
−1 4.19 (T1T ) at various pressures plotted as a function of temperature...... 55
4.20 Temperature and pressure dependence of charge disproportionation extracted
from T1 data...... 56
5.1 Field dependence of 13C NMR spectra at T = 0.749 K...... 60
5.2 Plot of temperature and field sweep for an in-plane magnetic field...... 61
−1 5.3 (T1T ) enhancement in the FFLO state under in-plane magnetic field..... 63
−1 5.4 Comparison of the field dependence in (T1T ) and the first moment...... 63
5.5 Angular dependence of spectra in the FFLO state at B = 9.25 T and T = 0.79 K. 64
−1 5.6 (T1T ) enhancement in the FFLO state under out-of-plane magnetic fields... 65
−1 5.7 Molecular specific (T1T ) analysis under out-of-plane magnetic fields...... 67
−1 5.8 Molecular specific (T1T ) analysis of an in-plane field B = 10.5 T...... 68
5.9 Sketch of density of states of a d-wave SC with nodes in the modulation of order parameter...... 70
−1 5.10 Temperature dependence of T2 at B = 10.5 T...... 72
ix −1 −1 5.11 Modeling of the effect of modulated order parameter on (T1T ) and T2 .... 74
x LIST OF TABLES
4.1 Diagonalized components of the hyperfine shift tensor for the four sites of ET molecules...... 45
4.2 Diagonalized components of the orbital shift tensor for ET molecules...... 45
xi ACKNOWLEDGMENTS
First of all, I would like to thank Stuart for not only showing me what a true physicist is, but also being a very kind advisor. Though I might not choose an academic career, I would want to become a scientist like you if I did. It was a great pleasure to work in your group. I’d like to thank Professor Yang, for all the support especially the first year when I joined UCLA. I’d also like to thank Professor Streit and Professor Goorsky to be on my committee.
When I first joined Stuart’s group, I knew nothing about experimental condense matter physics. Georgios really helped me a lot. You are probably one of the best teacher I knew. During my time at Dresden, Germany, Sebastian and Hannes were so helpful and kind. It was a great time working with you guys. At the stage when I spent a lot of time in the lab, lunch break was such a relaxing moment. I enjoy chatting with Alvin, Pedro and Erik about everything in life, and I will remember all the funny moments with Yongkang and Andrej, besides physics you guys shared with me. I want to thank Adam, for all the help in the lab when I cannot be there, and without Aaron and Teresa, my last year at UCLA would for sure be less fun. I’d like to thank everyone sincerely for your companion and help.
I want to thank all my friends at UCLA and Caltech. Our weekly volleyball time, hanging out, and friendship were an essential part for my Ph.D. time at Los Angeles. Lastly, I truly appreciate all the support from my parents, my sister Tina, my aunt Sue and my significant other Riva. Without their support, I may not complete my education.
xii VITA
2009 Bachelor of Science in Physics, National Taiwan University
2012 Master of Science in Optoelectronics, National Taiwan University
2011-2012 Visiting Researcher, University of California, Berkeley
2013-2018 Graduate Student Researcher, University of California, Los Angeles
2014-2018 Teaching Assistant, University of California, Los Angeles
xiii CHAPTER 1
Introduction
In this dissertation, we present the spectacular physics in a fully-organic charge transfer salt
β”-(BEDT-TTF)2SF5CH2CF2SO3 (β”-SC), where the charge transfer salts are compounds consisting of electron donor and acceptor molecules. β”-SC has a so-called quarter-filled band, one electron charge on every two donor molecules, and is a quasi-two-dimensional (q2D) material with a superconducting ground state. The low dimensional nature of this material makes it subject to electron correlation, and therefore, it is an ideal system for study- ing physics of correlated materials, which is an important topic in modern condensed matter physics. The topics covered in this dissertation include charge ordering in a quarter-filled two dimensional system (2D) and its relation to superconductivity, as well as the inhomo- geneous high field superconducting phase, the Fulde-Ferrell-Larkin-Ovchinnikovin (FFLO) phase. Introduced here are concepts that are necessary for understanding the phenomena under study.
1.1 Unconventional Superconductivity and Pairing Mechanism
Superconductivity was first discovered in 1911 by Heike Kamerlingh Onnes in mercury, who
observed drop in resistivity to zero below a critical temperature Tc. Zero resistivity, together with the Meissner effect, where magnetic field flux is expelled in the superconducting mate- rial, are two signatures of superconductivity. Ginzburg–Landau (GL) theory was proposed in 1950 to explain the phenomena based on Landau’s theory of second-order phase transitions. GL theory assumes that a complex order parameter describes the superconducting state. By expanding a free energy functional in terms of the superconducting order parameter, two
1 characteristic length scales λ and ξ naturally emerge. The penetration depth λ is the length scale associated with magnetic field penetration in superconductors (SCs), whereas the su- perconducting coherence length ξ describes the spatial extent or size of the superconducting pair. With these two characteristic length scales, GL theory describes macroscopic behav- iors of SCs under a finite temperature and magnetic field fairly well. A microscopic theory only emerged more than 40 years after Onnes’ discovery. In 1957, Bardeen, Cooper, and Schrieffer proposed a theory (BCS theory) which can quantitatively account for numerous characteristics of superconductivity [2]. BCS theory shows that when an attractive force between electrons exists, a superconducting gap opens up at Fermi surface at a sufficiently low temperature, and electrons pair in a phase coherent state. This type of electron pair was named Cooper pair after Cooper. In the original BCS theory, Cooper pairs are in a singlet state (the spin part of the wave function of electron pair is antisymmetric with zero total spin angular momentum), and the superconducting gap is isotropic in momentum space. The attractive interaction was found to originate in the coupling between electrons and phonons, and was experimentally verified by the effect of isotope substitution on the superconducting
transition temperature Tc. A naive but rather easily understood picture would be that the lattice distorts when an electron, which carries negative charge, travels by. This region of the lattice then temporarily becomes positively charged and attracts the next electron. This
electron–phonon coupling leads to the conventional SC where Tc scales linearly to the Debye
frequency ωd. Throughout this dissertation, the term conventional SC refers to SCs with isotropic superconducting gap and singlet Cooper pair pairing that is mediated by electronic coupling to the lattice.
1.1.1 Magnetically mediated paring? Cases of cuprates and a half-filled system
Another class of SCs is the unconventional SCs, for which the pairing cannot not be explained by electron–phonon coupling. Research interests were drawn to unconventional SCs not only because of the new physics involved, but also because of the higher Tc of some of the unconventional SCs that opens up the possibility for practical applications. For instance, the cuprate SCs exhibit Tc above liquid nitrogen temperature T = 77 K. Unconventional
2 SCs all exhibit physical properties associated with electrons that are strongly correlated, for which the Coulomb repulsion between electrons is strong. In the case of cuprate SCs, they typically have one or more superconducting layers of CuO2 separated by other cation species. Regarding the copper ions, the two s electrons are ionized, and therefore, the electronic configuration is 3d9. The partially filled d orbital of copper would naturally lead to a metallic state in conventional band theory. This is not the case when observing the so- called parent compound La2CuO4. Figure 1.1 presents a phase diagram of La2CuO4 under various temperature and doping conditions. La2CuO4 has an antiferromagnetic insulating ground state, and becomes superconducting with an appropriate amount of doping. In the case of hole doping, for example, this is accomplished by replacing La3+ ions with Sr2+ atoms. Later, it was understood that the magnetically insulating ground state of undoped
La2CuO4 can be classified as a Mott–Hubbard insulator, which is a class of material that should be conducting in traditional band theory, but is in fact insulating.
Figure 1.1: Phase diagram of La2CuO4 under electron/hole doping [3]
A Mott insulator can be understood as a strongly correlated electronic system and has been discussed in metal oxide materials long before the discovery of unconventional SCs. Mott insulators describe materials that have a half-filled band, and are expected to be metal- lic in conventional band theory but in fact has an insulating ground state. This phenomenon
3 originates from a strong interaction between electrons in the form of Coulomb repulsion. On the contrary, in conventional band theory, in the independent electron approximation, which assumes one may simply consider the picture of a single electron under potential created by ions and the effect of other electrons can be neglected, is commonly adopted. The Hubbard model below is widely believed to account for the essential physics of Mott insulators.
X + X H = t (ciσcjσ) + U (ni↑ni↓), (1.1)
+ where ciσ and ciσ are the creation and annihilation operators for electrons at site i with spin σ, respectively; ni↑ and ni↓ are the number operators for site i with spin up or down, respectively; t is the transfer integral which is a direct measure of bandwidth; and U is the onsite Coulomb repulsion. The transfer integral t corresponds to the kinetic energy gain related to electronic intersite hopping.
Here, we discuss an example of a half-filled system with square lattice to demonstrate how Coulomb repulsion can lead to a magnetic insulating ground state, as shown in Figure 1.2. Considering that each lattice site can be occupied by up to two electrons of spin up and down, a half-filled system will be metallic. However, in the limit of strong Coulomb repulsion between electrons, i.e. U → ∞, intersite hopping is reduced. The electrons are localized provided that U/t is sufficiently large. In the limit of U >> t, the t − J model can be derived from the Hubbard model. The indirect exchange interaction J between electrons
t2 J ∝ U
characterizes the nature of electron correlation. For J > 0 or J < 0, electron correlation is antiferromagnetic or ferromagnetic, respectively. In cuprates and many other unconventional SCs, this correlation term J > 0 leads to an antiferromagnetic ground state.
In Figure 1.1, as La atoms are substituted by Sr atoms in a hole doping scenario, more vacancies appear. When a nearby electron hops to a vacant site, its spin will be paral- lel to its neighboring electrons, and is not favorable energy-wise. Hole doping, therefore,
4 w/o correlation w/ correlation E E T Insulator U EF Metal
AF SC
DOS Band Diagram U/t
Figure 1.2: Effect of electron correlation on a half-filled system. Distribution of electron on lattice sites with (upper right) and without (upper left) electron correlation. The energy band diagram (lower left) and the phase diagram (lower right) of a correlated half-filled system
suppresses the antiferromagnetic ordering. When the doping level increases, the transition temperature from the paramagnetic state to antiferromagnetic state decreases and eventually reaches zero. If the doping level is further increased, a superconducting state emerges. The proximity of the magnetic ground state to the superconducting state triggered the idea that the unconventional superconductivity may be related to or even originated from magnetism. Moreover, different experimental techniques, including neutron scattering and nuclear mag- netic resonance (NMR), have revealed enhanced spin fluctuations above the superconducting
transition temperature Tc, indicating the possible relationship between superconductivity and magnetism [4–8]. Recently, reports have been published on the coexistence of charge ordering and charge fluctuation in underdoped cuprates, which has triggered research ac- tivities in the interplay between charge degrees of freedom and superconductivity [9–11].
5 Nevertheless, because of ubiquitous magnetic ordered states proximate to the superconduc- tivity among cuprate compounds and other unconventional SCs, magnetic origin remains a promising conjecture regarding the pairing mechanisms.
1.1.2 Charge ordering and superconductivity arising from electron correlation in a quarter-filled system
w/o correlation w/ correlation T
E E
U Metal
EF 3V CO
SC ? DOS V/t Band Diagram
Figure 1.3: Effect of electron correlation on a quarter-filled system. Distribution of electron on lattice sites with (upper right) and without (upper left) electron correlation. The energy band diagram (lower left) and the phase diagram (lower right) of a correlated quarter-filled system
New physics emerges when the system is not exactly half-filled, and charge degrees of freedom may need to be taken into consideration even when U/t → ∞. For example, consider a quarter-filled system using the Hubbard model. The system is expected to be metallic because other vacancies exist as shown in Figure 1.3. To depict materials with a quarter-filled band and an insulating ground state, an extra term in the Hubbard model is
6 Figure 1.4: Calculated phase diagram of a quarter-filled system on a q2D square lattice based on the extended Hubbard model. V , t, and T represent intersite Coulomb repulsion, the transfer integral, and temperature, respectively [12] included, namely intersite Coulomb repulsion. Here, we introduce the commonly adopted Extended Hubbard model [13, 14].
X + X X H = t (ciσcjσ) + U (ni↑ni↓) + V (niσnjσ0 ) (1.2)
Qualitatively, with sufficiently large U/t, and V/t, electrons tend to stay away from each other forming a periodic charge ordered structure, and the quarter-filled system is insulat- ing. This charge ordering differs from the charge density wave (CDW), which is associated with periodic lattice distortion and is mostly observed in one-dimensional (1D) materials. The charge ordering here, of the Extended Hubbard Model, arises from electron–electron correlation, whereas in the case of CDW, Fermi surface nesting and electron–lattice cou- pling are involved. Theoretical calculation has been performed on a square lattice of a
7 quarter-filled system using Extended Hubbard model with H¨uckel approximation, which is an approach commonly adopted that approximates molecular orbitals by superposition of atomic orbitals, and is historically used to determine the energy of π electrons [12, 15]. With a limit of U >> t, double occupancy is prohibited and simplified the calculation. The results
are depicted in Figure 1.4, which shows that when V/t exceeds a critical value (V/t)c, the system forms an insulating charge-ordered state in a checker board pattern. When V/t is
smaller than (V/t)c, the system is metallic with a homogenous distribution of charge. With the suppression of charge ordering, a quantum critical point exists, which is a point in the phase diagram where a continuous phase transition occurs at T = 0; in a mean field theory, it has been shown that in the limit of U → ∞, charge fluctuations near this critical point may lead to stabilization of a superconducting state with electron pairing function of d-wave symmetry [12], which describes the anisotropic superconducting gap as opposed to isotropic s-wave symmetry described in BCS theory as shown in Figure 1.5. Also, a cartoon of how magnetic (spin) or charge ordering can mediate superconductivity is shown in Figure 1.6.
s dx2−y2
Figure 1.5: An isotropic s-wave superconducting gap ∆ = ∆0 (left) and an anisotropic dx2−y2 superconducting gap ∆ = ∆0cos(2θ) (right) with a 2D cylindrical fermi surface, where θ is the angle in momentum space. Solid lines are the fermi surface and dash lines present the magnitude of superconducting gap ∆
Chemical and physical pressure are common tuning parameters used to study the effect of correlation by changing V/t. In a simple case, the effect of pressure is as follows. Hydrostatic
8 compressive pressure leads to the shrinking of lattice spacing. In the extended Hubbard model, the onsite Coulomb potential U is not affected by a change of lattice spacing, whereas
−1 the intersite Coulomb potential V would scale as V (r) ∝ r . For the transfer integral t, which presents the wave function overlap between sites would scale as t(r) ∝ e−r. Therefore, as lattice spacing decreases, changes in t would dominate, leading to a decreased V/t. In other words, as pressure increases, electrons become more delocalized, and consequently, the correlation effect becomes less important.
δ- δ-
δ+ δ+
Figure 1.6: Cartoon of spin- (left) and charge- (right) mediated pairing
1.2 A Novel High Field Superconducting Phase: FFLO State
Whereas temperature suppresses superconductivity as the energy scale of thermal fluctuation becomes significant compared with the superconducting condensation gap, it is not the only parameter does that. Magnetic fields can also destroy superconductivity. Based on the response under an applied magnetic field, SCs are usually classified according to two types.
Type I SCs expel the magnetic field for H < Hc, and the normal state is re-established via a first-order transition at greater field strength. Type II SCs have the lower and upper critical fields Hc1 and Hc2. For H < Hc1, SCs act as a Type I SC, expelling completely the magnetic flux. For Hc1 < H < Hc2, the magnetic field H penetrates the specimen hc in the form of vortices which possess quantized flux Φ0 = 2e and cores with vanishing superconducting order parameter amplitude. The vortices in Type II SCs are named after
9 theoretical physicist A. Abrikosov, who predicted them based on GL theory [16]. When H reaches Hc2, SCs undergo a second-order phase transition to the normal state. In GL theory, the ratio between penetration depth λ and superconducting coherence length ξ depicts the tendency to be Type I or II SCs. Qualitatively, for large λ and small ξ values, an interface between normal and superconducting regions has a negative surface energy and is more stable because the magnetic field penetrates deeper into the superconducting state instead of being concentrated only at the core, and SC pairing density recovers quickly to a bulk SC value as one moves away from the interface, retaining the SC condensation energy gain,
λ as well as vice versa. Quantitatively, when a parameter κ = ξ , which associates to the two length scales λ and ξ in GL theory, is smaller than √1 , an SC is Type I, and when κ is larger 2 than √1 , an SC is Type II [16]. 2
1.2.1 Orbital limit and Pauli limit of critical field
In Type II SCs, when the applied magnetic field is further increased above Hc1, the density of the vortices increases accordingly. When the density of the vortices has become so high that the vortices start to overlap, superconductivity is destroyed. Equivalently, from another perspective, the transition from superconducting state to normal state occurs when the kinetic energy of the screening current dominates over the SC condensation energy. This SC
orb pair breaking mechanism is called the orbital limit, wherein the critical magnetic field Hc is given by Φ Horb ' 0 , c 2πξ2
where φ and ξ are the quantized magnetic flux and coherence length, respectively. The orbital field can be estimated using the slope of phase boundary in the H −T phase diagram approaching Tc [17]. dH0 H = −0.7T c2 , c2 c dT
0 where Hc2 is the upper critical field at zero temperature.
An alternative pair breaking mechanism is the Pauli paramagnetic pair-breaking of super- conductivity, which relates to the electronic spin degree of freedom. The Pauli pair-breaking
10 mechanism originates from the Zeeman effect of quasiparticles under a magnetic field which leads to spin-split Fermi surface. In a metal, as is the case for conventional SCs, magnetic
2 susceptibility χp = µ0µBρ(EF). With an application of an external field H, the spin degen- eracy of quasiparticles is lifted and the spin-up and spin-down Fermi surface are split by
2 ∆E = ±χpH . Singlet SCs, which have antiparallel spins, suffer from this pair-breaking mechanism. This corresponding critical field of SCs is given by the Chandrasekhar–Clogston limit [18, 19] √ p 2∆ H2 = , (1.3) gµB where ∆ is the superconducting gap; g is the electron g-factor; and µB is the Bohr magneton. √ orb 2Hc2 To quantify which pair-breaking mechanism is dominant, the Maki parameter α ≡ p is Hc2 commonly defined [20]. For a large α > 1.8, the transition from SC state to normal state is Pauli-limited, and when α < 1.8, this transition is orbital-limited. In BCS theory, coherence length ξ = ~vF . Together with the above-mentioned expression, one can find that α ∝ ∆ . π∆ EF
As in most materials, EF is of the order of 1–10 eV and the superconducting gap ∆ is of the order of 1 meV. Therefore, most SCs are orbital-limited in terms of their critical magnetic field.
Figure 1.7: Illustration of modulated superconducting order parameter. The modulation of superconducting order parameter is presented by the sinusoidally oscillating bold curve. At the node of modulation, dark blue arrows are the unpaired electrons whereas at the maximum or minimum of modulation, light blue arrows with yellow circles represent the Cooper pairs
11 1.2.2 FFLO state and existing experimental evidence
In the case that the orbital effect is suppressed and the Pauli paramagnetic limit is the dominating pair breaking effect, a high field SC phase overcoming the Pauli limiting field p Hc2 was predicted independently by Fulde and Ferrell, and Larkin and Ovchinnikov in mid 60s, after whom this high field SC phase was named [21, 22]. In the FFLO phase at a field p above Hc2, Cooper pairs acquire finite momentum q in moment space, in contrast to the Cooper pairs with zero total momentum described in BCS theory. In real space, this finite momentum q results in spatial modulation of the superconducting order parameter ∆ in the form of ∆(r) ∝ cos(qr), where ∆ is the superconducting gap. This order parameter modulation is associated with real space nodes, in the gap as shown in Figure 1.7. Although the FFLO state loses energy reduction from quasiparticle condensation at the nodal region, it gains from the paramagnetism of normal state electrons. Therefore, by sacrificing part of p its volume to the normal state, superconductivity survives to higher fields above Hc2.
After the FFLO state was proposed, experimental studies, though mostly remain elu- sive, were only reported 30 years later, because of the stringent conditions required for the stabilization of the FFLO state. Observation of the FFLO state requires a Maki parameter α > 1.8, signifying the dominant Pauli paramagnetic limit effect, and also requires SCs to be in the clean limit, where the mean free path is much longer than the SC coherence length ξ, so that the characteristic length scale of modulation is well-defined [23, 24]. Heavy fermion com- pounds were proposed as candidates for FFLO states, supported by experimental evidence. Heavy fermion compounds with large effective mass—thus small fermi energy—lead to large
Maki parameters. For example, CeCoIn5 is reported to have an α of approximately 3.5 [25].
Several thermodynamic measurements of CeCoIn5 have revealed an additional distinct phase before it reaches normal state, which was interpreted as the transition from uniform SC state to FFLO state [26, 27]. Furthermore, a change from second- to first-order transition was observed at T *< Tc, which is not expected for uniform Type II SCs but predicted to occur for a paramagnetic limited SCs [20]. However, antiferromagnetic order was later observed in
CeCoIn5 by NMR and neutron scattering [28, 29]. The coexistence of magnetic ordering and
12 superconductivity is interesting, but indicates that different or additional physics is relevant.
Superconducting organic charge transfer salts, which comprised of alternating conducting and insulating layers, are ideal candidates for the FFLO state. Depending on the coupling between stacks of conducting molecules in the conducting layers, the compound can be quasi-1D or -2D. The highly anisotropic nature is the key to suppress the orbital pair- breaking effect previously described. In particular, for magnetic fields applied parallel to the conducting plane, penetration of magnetic flux occurs via Josephson vortices instead of Abrikosov vortices, which are observed commonly in Type II SCs. Unlike Abrikosov vortices, Josephson vortices are coreless in the sense that the cores are confined in the insulating layer, and thus do not destroy the SC pairing. In addition, these organic compounds are usually extremely clean with long mean free paths at the order of 100 nm and short superconducting coherence length of between 1 and 10 nm.
Altogether, these characteristics favoring formation of the FFLO state have triggered nu-
merous studies on these organic compounds. Quasi-one-dimensional compounds TMTSF2(X),
with X being PF6 or ClO4, have been reported with critical fields higher than the predicted p Hc2 although no further evidence of the FFLO state so far [30, 31]. On the other hand, by various experimental methods, the Q2D compound κ-(BEDT-TTF)2Cu(NCS)2 has been p found to show an upper critical field exceeding Hc2, and also an additional phase transi- p tion near to the expected magnetic field H ∼ Hc2, indicating the transition from a uniform superconducting state to the FFLO state [32–34]. To microscopically study the modu- lated order parameter, NMR stands out from the macroscopic thermodynamic methods. p Using NMR, an inhomogeneous spectral broadening at a field above Hc2 was observed in
κ-(BEDT-TTF)2Cu(NCS)2, consistent with a modulated superconducting order parameters p [35]. Moreover, a sharp enhancement in the NMR spin lattice relaxation rate above Hc2 was reported and interpreted as evidence for the existence of nodes of superconducting order pa- rameter modulation, in which normal-state quasiparticles reside and contribute to enhanced
relaxation [36]. However, detailed studies of κ-(BEDT-TTF)2Cu(NCS)2 are hindered by the requirement of high magnetic fields above B = 21 T to reach the possible FFLO state. Later, β”-SC, as another member of the clean and highly anisotropic organic SC family, attracted
13 p scientists’ attention because of its lower Bc2 ∼ 9.3 T, which is achievable in common lab magnets, and is, therefore, the material under study in this dissertation.
14 CHAPTER 2
Quarter-filled Quasi-two-dimensional Organic
Superconductor β”-(BEDT-TTF)2SF5CH2CF2SO3
2.1 Crystal Structure
β”-SC belongs to a group of organic charge transfer salts (BEDT-TTF)2X consisting of al- ternating layers of electron donor molecules bis(ethylenedithio)tetrathiafulvalene, commonly abbreviated as BEDT-TTF or ET, and various anions X. The ET molecule, as shown in Figure 2.1(d), is synthesized using organic methods. The β”-SC crystal is grown using an
electrocrystallization method, and has a superconducting transition temperature Tc of 5.2
K[37]. Every two ET donor molecules contribute −1e charge to the anion SF5CH2CF2SO3, resulting in +0.5e per ET molecule; hence, a nominally quarter-filled band.1 Different an- ions can be utilized to tune the distance between ET molecules, which is a method of chemically tuning pressure, and can also affect the packing motif of ET molecules; patterns are designated using Greek letters, such as β, κ, θ, α. Figure 2.1 presents a typical crystal structure and examples of packing arrangements. Some result in dimerized arrangement of ET molecules, such as the κ compound, which is not the case for the β” family. The ET molecules stack along the a axis and ET/anion layers alternate along the c axis. With larger transfer integrals along the interstack b direction than along the a axis, and therefore higher conductivity, synthesized crystals are often longer along the b axis because they grow faster. The crystal system of β”-SC is triclinic with unit cell parameters of a=9.1536 A,˚ b=11.4395 A,˚ c=17.4905 A,˚ α=94.316◦, β=91.129◦, γ=102.764◦ at T = 123 K and a=9.260 A,˚ b=11.635
1Here, a quarter-fill band only refers to the fact that every two ET molecules share +1e charge. The Brillouin zone is, in fact, half filled.
15 A,˚ c=17.572 A,˚ α=94.69◦, β=91.70◦, and γ=103.10◦ at T = 298 K [37]. From T = 298 K to 123 K, the a and b axes contract around 1.2% and 1.7%, larger than the change in c axis, which contracts by approximately 0.5%. Four ET molecules exist in a unit cell, with two sets being crystallographically inequivalent. One set of ET molecules sits closer to the SO3 ligand and the other sits closer to the SF5 ligand.
The HOMO energy of ET molecules was calculated using the Extended H¨uckel method, and was demonstrated to mainly be composed of pz orbitals. Moreover, nearly 25% of the charge in the HOMO resides in the central carbon nuclei and 42% resides on the nearby sulfur nuclei. By contrast, protons, which are well known NMR active nuclei with high sensitivity at the two terminals of ET molecules, have a nearly zero population density of conduction electrons based on the calculation. Because both the NMR active nuclei 13C and 33S have low natural abundance of approximately 1%, the central two carbon nuclei are therefore substituted with 13C to probe hyperfine coupling and shifts for our NMR study, as shown in Figure 2.1(d) with the two 13C sites being structurally inequivalent in ET compounds typically because of their different distances to the anion layer.
2.2 Magnetic, Electrical and Optical Properties in Normal State
Band calculations of β”-SC describe two Fermi surfaces, consisting of a 2D hole pocket and a pair of 1D sheets [37, 38]. Although the quantum oscillation measurement did not ”see” the 1D Fermi surface as predicted in the calculation, a 2D Fermi surface consistent with the calculation was observed, as shown in Figure 2.2[38]. The 2D Fermi surface was found to be noncircular, with an area that was approximately 5% of the first Brillouin zone. This is much smaller than the expected 25% in the calculation. Furthermore, cyclotron electron effective mass was evaluated to be mc ∼ 1.9me. Both pieces of evidence indicated localized electrons and the crucial role of electron correlation in β”-SC. Resistivity of β”-SC showed a
monotonic decrease from room temperature to Tc with a quadratic temperature dependence at low temperature, measured by the four-probe method [1, 39]. The relative magnetic susceptibility measured by electron spin resonance (ESR) showed an enhancement at higher
16 (a) (b)
anions
BEDT-TTFs
β" θ κ anions
(c) (d) b
c a H S C12 C13 b
a
Figure 2.1: (a) Crystal structure of (BEDT-TTF)2X compounds with alternating layers of ET molecules and anions; (b) examples of packing motif in (BEDT-TTF)2X compounds, looking along the long axis of ET molecules; (c) packing arrangement of ET molecules in β”-SC viewed along the c axis; and (d) structure of ET molecules with two central carbon sites substituted with 13C nuclei
temperatures, whereas it was constant at temperatures below 100 K in accordance with Pauli
susceptibility in metallic compounds [40]. All in all, above Tc, β”-SC shows metallic behavior. The resistivity anisotropy is reported to be in the range of ρ⊥ ∼ 100 − 1500, whereas the ρk −1 in-plane resistivity is in the order of 1 (Ωcm) [39, 41], and ρ⊥ and ρk represent out-of- plane and in-plane resistivity, respectively. In the a-b plane, slight anisotropy in microwave
conductivity of σb ∼ 1.35 was extracted from ESR line shape analysis. σa Evidence for charge ordering and fluctuations in β”-SC have been reported in optical measurements. In the infrared reflectance study, a charge sensitive vibration mode of ET molecule near 2500 cm−1 shows two peaks for T < 200 K, demonstrating charge dispropor- tionation of ∆ρ ∼ 0.2e locally [42]. In accordance with the later transport measurement in [1], infrared reflectance of β”-SC shows weak metallic characteristics across the whole tem- perature range [43]. An extremely narrow Drude peak was observed at lower frequencies in a later experiment at T < 200 K, which is typical for organic conductors [42]. In the same re-
17 Figure 2.2: Angular dependence of oscillation frequency F and effective cyclotron mass µc (inset) in Shubnikov–de Haas measurement (left) and a calculated band showing a 2D hole pocket and a pair of 1D electron sheets (right) [38] port, a broad band in the mid-infrared range of optical conductivity is interpreted as charge ordering with charge fluctuations between sites, in accordance with numerical calculation using Extended Hubbard Model in [44], and an intense peak at 300 cm−1 upon cooling is interpreted as a charge fluctuation band related to the transition between itinerant electrons at the Fermi surface and relatively localized electrons in the charge-ordered phase. Simi- larly, in microwave measurement, an increased resistivity at high frequency (approximately 30 GHz) was reported for T < 30 K, which was also attributed to charge fluctuation [39]. All these reports in β”-SC indicate possible interplay between superconductivity and charge fluctuation.
2.3 Superconducting Properties
In the superconducting state, β”-SC exhibits strong anisotropy in the upper critical field, which is expected for layered SCs because magnetic flux penetrates through the insulating layer to form Josephson vortices when the magnetic field is aligned to the conducting plane
(Section 1.2.1). At T ∼ 2 K, out-of-plane critical field (Bc2⊥) and in-plane critical field
(Bc2k) are shown to be approximately 1.4 T and 10 T, respectively, by thermal expansion and specific heat measurement [45, 46]. The temperature dependence of specific heat at zero
field allows the extraction of the superconducting gap ∆0 assuming s-wave SC, and indicates
18 an SC described by BCS theory in the strong coupling limit, wherein the coupling param-
eter ∆0 ∼2.15 [46, 47]. Although specific heat measurements are interpreted as evidence kBTc consistent with s-wave SC, the symmetry of the gap is still under debate, similar to κ com- pounds as other experimental probes commonly show a d-wave nature of the superconducting
gap. Substitution with deuterium on ET molecules demonstrated a suppression of Tc, and was attributed to shortened bond length of the C–D bond and longer distance between ET
molecules [48]. Specifically, Tc increases with deuterium substitution. This is in contrast to what BCS theory predicts because the electron–phonon coupling strength is weakened with heavier atoms. In-plane transport measurement showed a four-fold symmetry, which
agrees with the dx2−y2 superconducting gap symmetry, as predicted in former calculations of a charge fluctuation induced superconducting state [12, 49].
The out-of-plane and in-plane superconducting penetration depth are reported to be
λ⊥ ∼ 800 µm and λk ∼ 1 − 2 µm, respectively. Because the interlayer spacing s of β”-SC is
∼ 17.5 A,˚ λk ∼ 2 µm would result in retaining 99.95% of the applied magnetic field in the conducting layer, making NMR measurement possible. The anisotropy parameter γ is defined
λ⊥ as γ = ∼ 400, indicating the 2D nature of β”-SC. The coherence lengths ξ⊥ and ξk were λk also extracted from upper critical field inferred from thermal expansion measurements to be
7.9 A˚ and 144 A,˚ respectively, exhibiting high anisotropy as well. A ξ⊥ value smaller than interlayer spacing s indicates no superconducting pair wave function overlap between layers
[45]. Moreover, with the mean free path l estimated to be approximately 100 nm, l >> ξk reveals the clean nature of β”-SC[46]. Most crucially, angular-dependent magnetoresistance measurement detected no evidence for the existence of coherent transport between adjacent conducting layers, which is a clear cut evidence of q2D characteristics [50, 51].
All the abovementioned studies illustrate that β”-SC is an ideal candidate for stabiliza- tion and study of the inhomogeneous FFLO state. Indeed, several experiments have been p interpreted as evidence for an FFLO phase. The Pauli-limiting field Hc2 can be estimated from Equation 1.3. Utilizing the coupling strength parameter α extracted from specific heat p measurement together with Tc, µ0Hc2 is estimated to be ∼9.73 T. In some earlier measure- ments of β”-SC, the upper critical fields above the Pauli-limiting field have already been
19 reported [45, 52]. Moreover, an upturn of Hc2 was reported at T ∼ 1.7 K in later specific heat measurements, which researchers found to be very sensitive to applied magnetic field orientation, as expected for the FFLO state [46]. The order of phase transition in this study from FFLO to normal state was interpreted as a second-order phase transition. Whereas the temperature sweep in specific heat measurement is not sensitive to the phase boundary of the uniform superconducting state to FFLO state because this boundary barely varies in temperature, a magnetic field sweep in specific heat measurement reveals a hysteric first- order transition at B = 9.1±0.5 T [53]. A further microscopic study previously performed in my group reported probing the modulation of the superconducting order parameter directly using the NMR spectrum. Spectral modeling indicated that the line shape is consistent with a 1D modulation of the superconducting order parameter, agreeing with a highly anisotropic 2D Fermi surface of β”-SC [54, 55].
Figure 2.3: Phase diagram of β”-SC extracted from specific heat measurements; Tc values at different angles are plotted in the inset where T * corresponds to an anomaly in specific heat that appears when the applied magnetic field is misaligned [46] .
20 CHAPTER 3
Experimental Technique
NMR is a powerful tool commonly utilized to study material properties through the lo- cal interaction of nuclear spins with their environment. NMR has various applications in chemistry, the petroleum industry, medical imaging, and other fields. In condensed matter physics, NMR serves as a noninvasive, microscopic tool for investigating the electron corre- lation of local environment in materials. In the following subsections, the basic principles of NMR and how nuclear spins respond to the electronic environment are introduced.
3.1 Response of Isolated Nuclei to a Magnetic Field
A nucleus with spin angular momentum I has a magnetic moment of
µn = γ~I, where γ and ~ are the gyromagnetic ratio and the Planck constant, respectively. When the nucleus is placed under magnetic field H0 taken as parallel to the z axis, the Zeeman effect lifts the degeneracy of nuclear energy levels. The energy levels of the nuclear spin in an applied field are given by
E = −µn · H0 = −γ~mzH0, where mz = −I, −I +1, ..., I is the eigenvalue of I. To probe the energy levels, an oscillating perturbative magnetic field H1 perpendicular to H0 is commonly used to excite nuclei. The selection rule then allows transitions between adjacent energy levels; that is, ∆mz = ±1.
21 The energy difference is equal to
∆E = γ~H0 = ~ω0.
Here, ω0 is known as the Larmor frequency and is typically in the range of 1–500 MHz in NMR experiments.
In a classical picture, which turns out to be useful for understanding NMR, nuclei are considered to have magnetic moment µn under external field H0 = H0zˆ in a laboratory reference frame. Magnetic field exerts a torque τ on µn where
dL dµ τ = = n = µ × H . dt γdt n 0
From the equation, if the magnetic moment is not parallel to the magnetic field, the magnetic moment will precess around the field. This is called Larmor precession. In a reference frame rotating at angular frequency ωref, the time derivative of a vector A is
dA dA ( ) = ( ) + (ω × A). dt lab dt rot ref
Therefore, dµ dµ ( n ) = ( n ) + (ω × µ ) = γµ × H . dt lab dt rot ref n n 0
In the rotating frame, dµ ( n ) = µ × (γH + ω ). dt rot n 0 ref
Therefore, if ωref is chosen properly so that γH0 + ωref = 0, the nuclear magnetic mo- ment is static in the rotating frame. Now, consider an oscillating perturbing field H1 =
H1cos(ωpertt)ˆx in the lab frame, which can be written as a combination of two fields rotating about z-axis with opposite rotating directions but same angular frequency ωpert. Taking
ωref = −ω0, then
dµ ( n ) = µ × (γ(H + H ) + ω ) = µ × γH . dt rot n 0 1 ref n 1
22 0 Therefore, in the rotating frame, this oscillating field H1 can be treated as a static field H1xˆ , about which the nuclear magnetic moment rotates, a phenomenon known as Rabi oscillations. Once the nuclear magnetic moment is not parallel to the z axis, Larmor precession begins.
In practice, this oscillating H1 field is created by a coil surrounding the sample to ”knock down” nuclear magnetic moment, and hence initiate Larmor precession. Simultaneously, this coil is used to collect voltage induced by Larmor precession, which is in fact the NMR signal recorded.
3.2 How Nuclei Interacts with the Environment
When the nucleus is not isolated, the Hamiltonian can be written as
H = Hzeeman + He-n + Hn-n + HQ = γ~I · (H0 + Hint) + HQ.
Hzeeman corresponds to the applied external field H0. He-n and Hn-n are the terms leading to modification of Zeeman splitting and are associated with interactions between nucleus and neighboring electrons, and nucleus and neighboring nucleus, respectively. Together He-n and Hn-n present the effect of an internal magnetic field Hint in the material. HQ is the quadrupolar term that accounts for the coupling of quadrupole moment Q to the electric
1 field gradient where Q exists in nuclei of I > 2 . Reported in this dissertation are results of 2 13 2 13 H and C; HQ is relevant for H but not C.
The He-n term can be written as
µ · µ µ · µ 3(µ · r)(µ · r) 8π H = [ L n ] + [ e n − e n ] − [ µ · µ δ(r)], e-n r3 r3 r5 3 e n
where µe is electron spin magnetic moment and µL is the electron orbital magnetic moment. In general, the first term describing magnetic field produced by electron orbital motion is quenched by the crystal field, and is only pronounced when a magnetic field is applied. The nuclear atomic number, which relates to the density of electron, is a major factor that affects this term, and the consequent change in local field is commonly referred to the orbital shift.
23 mz=+1/2
~ħω
mz=-1/2
H0=0 H0>0 H0+Hint
1 Figure 3.1: Energy diagram of a spin 2 nucleus under a magnetic field
For 13C, orbital shift is usually around 1–100 ppm. The anisotropic second term is the coupling of nuclei to electronic spin dipolar field and the isotropic third term corresponds to the onsite magnetic field mainly contributed by s-electron density at the nucleus, which is also called the Fermi contact interaction. These two terms describe the hyperfine interaction between electron spins and nuclear spins, and the change in local magnetic field from these two terms normalized to applied field, Knight shift, can be much larger than orbital shift.
The Hn-n term describing the dipolar interaction between nuclear magnetic moments can be written as µ · µ 0 3(µ · r)(µ 0 · r) H = n n − n n n-n r3 r5
where µn0 is the nuclear moment from neighboring nuclei. This nuclear dipolar interaction does not depend on the magnetic field; it only depends on the species of nucleus, and its distance and direction from other nuclei. For two 13C nuclei separated by approximately 1 A,˚ this interaction can cause a maximum change of approximately 5 kHz in frequency, and therefore, it can broaden or split peaks in the NMR spectrum.
24 3.3 NMR Observables
As previously described, NMR plays a critical role in condensed matter physics because of its ability to study the electronic environment locally. The utility of NMR as a micro- scopic probe can be demonstrated using the following example in Figure 3.2, where NMR spectra of different magnetic ordering are shown. Normally, when studying with a macro- scopic tool, such as when measuring magnetic susceptibility using a magnetometer made of a superconducting quantum interference device (SQUID), the magnetic susceptibility for the antiferromagnetic and paramagnetic phases are not straight forward to distinguish be- cause the susceptibility of both phases is small. In that case, careful quantitative analysis or temperature dependent measurement may be required. However, with the NMR spectrum, not only can it differentiate between antiferromagnetic and paramagnetic phases, but also can be used to infer the order parameter or, specifically, how the antiferromagnetic moment is aligned and distributed. While NMR spectrum reflects the static local field distribution,
−1 the NMR spin lattice relaxation rate T1 is the relaxation of nuclear spin mediated via the dynamic electronic environment. Below, we briefly introduce these two static and dynamic aspects of NMR measurement.
Phenomenologically, the local field Hint induced by the external field H0 results in a shift as follows. Note that the local field may exist by itself without applying the external
field H0 and can be much larger than H0 in a magnetically ordered phase.
H = H0 · (1¯ + K¯s + K¯orb)
where 1¯, K¯s, and K¯orb are identity, hyperfine shift, and orbital shift tensors, respectively.
In a metal, K¯s is given by
K¯s = A¯χs
where A¯ and χs are the hyperfine coupling tensor and spin susceptibility, respectively. Note that here, χs is taken to be isotropic but in general, it should also be written as a tensor.
As discussed previously, K¯s can be further separated into isotropic and anisotropic parts, as
25 Ferromagnetism Antiferromagnetic Paramagnetic
�Hint H0
ω0= �H0 ω0 ω0 Static NMR measurement – Spectra Sensitive to local magnetic field
Figure 3.2: Schematics of NMR spectra of different magnetic ordering showing the micro- scopic nature of NMR spectra, where γ, H0, Hint are the nuclear gyromagnetic ratio, applied external magnetic field and internal magnetic field, respectively
shown below [56, 57]. 8π Kiso = <| φ(0) |2> χ s 3 EF s 2 1 Kan = < > (3cos2α − 1)χ s 5 r3 s
2 Here, <| φ(0) | >EF is the average squared amplitude of Bloch wave function over the Fermi 13 surface. The anisotropic representation applies to onsite pz orbital, which is the case for C
NMR study on β”-SC. α is the angle between the applied field and z axis of pz orbital; and
χs scales with the density of states at the Fermi level ρ(EF).
−1 Furthermore, the spin lattice relaxation rate T1 describes how fast the nuclear magnetic −1 moment recovers back to thermal equilibrium after being perturbed. (T1T ) is given as follows [58]: X χ”(q, ω) (T T )−1 ∝ [A ], 1 q ω q ~
where Aq represents the hyperfine coupling perpendicular to the direction of the applied
26 field, and χ”(q, ω) is the imaginary part of dynamic susceptibility at wave vector q and NMR frequency ω. In a metal, the isotropic term, for which Fermi contact interaction is dominant, or the anisotropic term, for which a dipolar interaction from pz orbital is dominant, can be written as follows [56, 57]:
1 iso 64 = π3 3γ2γ2[<| φ(0) |2>]2 ρ2(E )k ~ e n EF F B T1T 9
an 1 π 3 2 2 2 4 1 2 2 = ~ γe γn(2 + 3sin α)( < 3 >) ρ (EF)kB T1T 6 5 r
where kB is the Boltzmann constant. Due to the anisotropic wave function of pz orbital,
−1 Ks and (T1T ) would have different angular dependences. For the isotropic Fermi contact
1 2 2 term, as χs = 2 ~ γe ρ(EF), the ratio
1 4πkB γe 2 2 = ( ) κ (T1T )Ks ~ γn
is a constant, which is the famous Korringa law. κ is the Korringa ratio and κ = 1 for Fermi liquid. In the presence of electron correlation, κ > 1 or κ < 1 for antiferromagnetic or ferromagnetic correlation, respectively.
27 CHAPTER 4
Result I - Charge Ordering and Superconductivity
β”-SC, as a quarter-filled organic SC, is an ideal candidate for studying the interplay of charge ordering, superconductivity, and the role of electron correlation. Here, we perform 13C and 2H NMR on β”-SC to study the superconducting state and relevant normal state properties to elucidate non-magnetic-origin unconventional superconductivity as proposed in [15] and discussed in Chapter 1. We start by introducing the methodology of magnetic field alignment in Section 4.1. In Section 4.2, the NMR spectra and spin lattice relaxation are measured between 1.5 and 300 K and interpreted as evidence of a novel non-magnetic- fluctuation mediated SC with charge-ordered normal state. In Section 4.3 and 4.4, electron correlation is revealed by Korringa ratio analysis and connected to the molecular dynamics at T > 100 K. Finally, in Section 4.5, broader aspects of the phase diagram of β” family are probed by high pressure NMR.
4.1 Applied Magnetic Field Alignment
For β”-SC, as a q2D SC composed of alternating conducting and insulating layers, a known approach to suppress the orbital effect on the superconducting critical magnetic field is by aligning the magnetic field to the conducting plane, as discussed in Section 1.2.1. Therefore, in β”-SC, it is possible to study the FFLO state that only appears when the superconductivity is Pauli-limited. Moreover, the signal intensity of NMR is enhanced at higher magnetic fields. Consequently, being able to examine the superconducting state at a higher magnetic field is beneficial and makes the experiment more efficient. For β”-SC, for example, the critical magnetic field Bc2⊥(T = 0 K) is approximately only 1.4 T for a field perpendicular to the
28 a-b plane, whereas Bc2k > 9 T [46].
To align magnetic field H, we probe the change of absorption in the tank circuit we built for NMR measurements, where the sample is placed in the inductance component, the coil. When the magnetic field is misaligned, a combination of Meissner effect and vortex pining effect comes into play and changes the inductance L of the coil. The resonance frequency
q 1 ω ∝ LC , where C is the capacitance in the circuit. Therefore, the change in L will lead to a shift of the absorption peak. By examining the absorption at a given carrier frequency as a function of field angle, the magnetic field can be aligned to the conducting plane. In practice, the carrier frequency is usually set at one side of the absorption peak so that the slope of absorption/frequency is the largest and most sensitive to any minor change of resonance condition. In Figure 4.1, the RF reflection of the tank circuit is observed at T = 1.7 K, which is well below Tc ∼ 5.2 K [37], and the features are expected to be symmetrical around the aligned angle. This method serves as a ”quick and dirty” method of roughly determining the orientation to an accuracy of ±1◦.1
To have a better precision, NMR measurements are utilized. Abrikosov vortices and quasiparticles in their cores are created when the magnetic field is not perfectly aligned to
−1 the conducting plane. These quasiparticles contribute to the spin lattice relaxation rate T1 −1 in our NMR measurement. An angular dependence of T1 can then reveal the existence of −1 Abrikosov vortices. In practice, the T1 measurement is conducted after a quick angular sweep of NMR intensity to determine the appropirate range of angle for the experiement due to a long T1 time scale in the superconducting state. In Figure 4.2, the angular de- −1 pendence of T1 is measured. The relaxation rate is minimized at the aligned condition, as Abrikosov vortices are suppressed. To the right of Figure 4.2, an NMR spectra comparison is presented between aligned and 1◦ misalignment to demonstrate the importance of accurate field alignment. The spectrum is significantly broadened when misaligned by merely 1◦ as
−1 the superconducting phase is destroyed. The accuracy of alignment using T1 is determined to be approximately ±0.05◦.
1The low accuracy is due to both hysteresis at low field and smeared feature at high field.
29 Matching(dB)
Angle (degree)
Figure 4.1: RF absorption measurement at T = 1.7 K at various fields; solid and dashed lines represent rotation in various directions, from which the field alignment to the conducting plane is determined by symmetry
4.2 13C NMR Spectra and Spin Lattice Relaxation
To understand the 13C NMR spectrum of β”-SC, we first examine the spectrum at T = 78 K, as shown in Figure 4.3.2 After field alignment, the field is aligned in the a − b plane precisely. Due to the crystal morphology, we estimate the orientation to have the crystal aligned along the a axis to within 5◦∼10◦.3 β”-SC has two inequivalent ET molecules, and each has two crystallographically inequivalent 13C sites. The two 13C nuclei on the carbon double bond are coupled to each other through nuclear dipolar coupling, and the resulting nuclear dipolar split is of the order of 0 ∼ 5 kHz. There are, therefore, at most eight peaks in total. The peaks at higher frequency, and hence larger shift, are assigned to the ”red”
2Main portion of the data in Section 4.2 are obtained by G. Koutroulakis et. al. originally and included here for completeness [59].
3A precise orientation of the magnetic field can be determined from NMR spectrum after extracting the hyperfine coupling tensor in Section 4.3
30 Aligned .) 1 - a.u ) K∙sec (
1 Misaligned 1° - ) Intensity ( Intensity T 1 T (
Frequency(kHz) Spectra Comparison Angle (degree) at T=750mK, B=9.25T
−1 Figure 4.2: Angular dependence of T1 is measured at T = 0.750 K, B = 9.25 T to align the magnetic field to the conducting plane (left); example of NMR spectra under aligned and off-aligned conditions, showing a significant difference by 1◦ misalignment (right)
− molecules located closer to the SO3 ligand and peaks at lower frequencies are assigned to the ”orange” molecules located closer to the SF5 ligand. For two dipolar-coupled nuclei with different Knight shifts, a quartet appears in the NMR spectrum and the NMR shifts of the quartet are given by the following equations [60–62].
1 1 K = K¯ − D − p4(∆K)2 + D2 + K a 2 4 orb
1 1 K = K¯ − D + p4(∆K)2 + D2 + K b 2 4 orb 1 1 K = K¯ + D − p4(∆K)2 + D2 + K c 2 4 orb 1 1 K = K¯ + D + p4(∆K)2 + D2 + K , d 2 4 orb
¯ Kin+Kout where K ≡ 2 and ∆K ≡ Kout − Kin. D is the nuclei dipolar coupling in shift with
γ~ 2 D = 3 (1 − cos θ), (4.1) H0r
31 γ MHz 13 ˚ where 2π = 10.7054 T is gyromagnetic ratio of C; r ∼ 1.369 A is the distance between two 13C sites; and θ is the angle between the magnetic field and carbon double bond. Note that the dipolar splitting D is a useful probe to help determine the in-plane orientation, which will be shown later in Section 4.3. The intensities of the four peaks are given by
Ia,d = 1 + sin2Θ
Ib,c = 1 − sin2Θ,
where sin2Θ = 1 − √ 0.5D . Using the quartet equation, the spectrum can be fit (∆K)2+(0.5D)2 well with a line width approximately 1 kHz, indicating that the NMR spectrum with site
dependent T1 is fully recovered.
Dipolar Splitting ( 2 + 2 )x2=8
2 different molecules with 2 inequivalent site on each
H S C12C13
Figure 4.3: 13C NMR spectrum obtained at T = 78 K, B = 8.4530 T for field in the a − b plane, roughly along the a axis (left); molecule model showing two inequivalent ET molecules, orange and red; also shown is a single ET molecule substitutes with 13C at the central carbon double bond (right)
A typical NMR spin lattice relaxation measurement is presented in Figure 4.4. Orange, red, and black curves represent the recovery curves for orange, red, and both molecules, respectively. For most T1 data reported in this dissertation, the saturation recovery method π π is used, where one or more 2 pulses is applied to saturate the nuclear spins, and a 2 ”read” π pulse is applied after different delay times. The 2 pulse is applied for ∼ 3 µs or less to have a -3 dB bandwidth larger than 150 kHz to cover the whole spectrum. Spectral data
are extracted for the points with longest delay time in T1 measurement, to ensure they are fully recovered. For the four inequivalent 13C sites, although they may have different
32 spin temperatures, T1 data reported in this dissertation are estimated by integrating across the whole spectrum. A stretched exponential equation shown below is used to obtain a characteristic time scale for the relaxation.4
−t β ( T ) I(t) = I0(1 − e 1 ),
where I0 is an arbitrary constant, and β is the stretched exponent which represents a distri- bution of T1.
1.2
1 NMR T1 relaxation
.) 0.8 a.u 0.6
0.4 Intensity (a.u.) Intensity ( Intensity 0.2
0
-0.2 10-1 100 101 102 103 Time(sec)Time(sec)
Figure 4.4: Representative recovery curve of T1 measurement; the orange, red, and black curves are the recovery curves for orange, red, and both molecules; dashed lines are the stretched exponential fit to the data
4 −1 In principle, a careful analysis can be done to estimate an average of T1 by integrating multiple −1 frequency regions and obtain a weighted average. Nevertheless, we have compared site-dependent T1 −1 versus mean (T1T ) extracted from the whole spectrum, and they share basically the same characteristics, indicating the validity of our estimate.
33 4.2.1 Normal state properties: a charge-ordered metal
13 We start by examining the C NMR spectra and spin lattice relaxation rate T1 of β”-SC above Tc. The measurements are conducted at B = 6.87 T in a temperature range of
−1 1.5∼300 K using a Variable Temperature Insert. In Figure 4.5(a), (T1T ) are plotted as a function of temperature. For a metallic environment, or more specifically, when weakly
−1 interacting electrons can be described by Fermi liquid theory, T1 scales with temperature
T and the square of the spin susceptibility χs, which is independent of temperature for a
−1 metal. Therefore, (T1T ) is expected to be constant for a metal, unlike, for example, the case of some magnetically ordered phases of which the spin susceptibility has temperature
−1 dependence. For Tc < T < 100 K, (T1T ) is nearly constant with very small temperature
−1 dependences, showing typical Fermi liquid characteristics. For T > 100 K, (T1T ) deviates from the constant and enhanced by approximately 5 times at 300 K. The enhancement above 100 K is believed to be indirectly related to the motion of the ethylene groups at the two ends of ET molecules, as will be discussed in detail in Section 4.4. Although differences
−1 exist between (T1T ) values of orange and red ET molecules, the overall temperature-
−1 dependent behavior is similar to the average (T1T ) integrated through the whole spectrum, as discussed previously.
(a) (b) 600 400 K K
500 �K � 1 - ) -1 (ppm) 300 10
K 400 K∙sec ( 300 200 1 - (ppm) ) T
1 200 Shift (ppm) T
( 100 Difference in Shift Shift Differencein Average Shift Shift Average 100 10-2 0 0 101 102 50 100 150 200 250 300 TemTemp pererature atur e ((K K)) Temp pererature atur e (K K))
−1 ¯ Figure 4.5: (a) (T1T ) for red, orange, and both molecules (b) average shift K = Kin +Kout and the difference in shift ∆K = Kout − Kin of the orange and red molecules plotted as a function of temperature
−1 Differences in (T1T ) between the orange and red molecules are further analyzed to study
34 charge disproportionation. For the ET molecules, the HOMO was calculated, which showed a high spin density near the central carbon double bond, as mentioned in Chapter 2 [63, 64]. 13C nuclei residing at the carbon double bond are hyperfine-coupled to the quasiparticles near the Fermi level, and are highly sensitive to the local spin density. Although we are not able to determine the net charge on each molecule, we can extract the relevant charge ratio on the two inequivalent molecules by assuming the following. First, the 3 by 3 hyperfine coupling tensor A is assumed to be proportional to the hole density on ET molecules. Second, the main contribution to the spin lattice relaxation is through the hyperfine coupling. This argument is supported by an estimate of the negligible contribution to spin lattice relaxation from motion of the terminal ethylene groups, which will be presented in Section 4.4. Therefore, the relaxation is mainly contributed by hyperfine interaction with quasiparticles. With this in mind, in Figure 4.6 we evaluate the temperature dependence of
s orange T1 hole density on red molecule red = . T1 hole density on orange molecule
−1 2 In a metal, (T1) ∝ A , where A =u ˆ · A · uˆ is the hyperfine coupling constant in a q orange T1 given field orientationu ˆ. red thus represents the hole density ratio between red and T1 orange molecules assuming absence of correlation between molecules [65]. This charge ratio between red and orange molecules increases monotonically from 1.4 to 2.1 (±0.1) as the temperature decreases from 300 K to 5 K, forming a metallic state with charge ordered stripes oriented along the a axis, which is the stacking direction for ET molecules. Unlike many other organic charge transfer salts in quarter-filled systems with charge disproportionation characteristics, such as α or θ compounds that exhibit metal-insulator transition at low temperature, β”-SC remains metallic across the whole temperature range [66–68]. The strong temperature dependence of charge disproportionation indicates the importance of electron correlations (onsite Coulomb repulsion U, and intersite Coulomb repulsion V ), rather than purely originating from asymmetric polar anions.
How is this charge ordering reflected in the NMR shift? First, we examine the tempera- ture dependence of the mean NMR shift K¯ of orange and red molecules, as plotted in Figure
35 2.2
2
0.5 1.8
1.6 Ratio) 1 1.4 T ( 1.2
1 0 100 200 300 Temperature (K)
Figure 4.6: Temperature dependence of intramolecule (orange and red square) and inter- molecule (blue square) charge disproportionation of red and orange molecules extracted from T1 data
4.5(b). NMR shift K can be written as K(T ) = Ks(T ) + Korb, where Ks(T ) and Korb are Knight shift and orbital shift, respectively. Thus, one would have to subtract the orbital shift to analyze the Knight shift, which reflects the local spin susceptibility. A common approach utilized for extracting Korb is the K − χ plot. Since the Knight shift Ks = Aχs, we can further write NMR shift K as K = Ks + Korb = Aχs + Korb. Accordingly, we can extract the hyperfine coupling constant A and orbital shift Korb from the slope and the intercept with the y axis. The macroscopic susceptibility was measured by SQUID and ESR, as shown in Figure 4.7(a) [40, 69]. The solid line is an interpolation to the experimental data to estimate
χs(T ). Notably, the susceptibility decreases by approximately 50% from T = 300 to 100 K and becomes nearly a constant below 100 K, showing the Pauli susceptibility of a metallic
−1 phase which also agrees with our (T1T ) analysis described previously. Furthermore, the average shift K¯ of orange and red molecules drop simultaneously upon cooling, qualitatively agreeing with decreased susceptibility, and is analyzed thoroughly below. A K − χ plot is
36 shown in Figure 4.7(b). At first glance, the results are abnormal and contradict our afore- mentioned charge disproportionation analysis. This is because the slope (hyperfine coupling constant A), representing charge density, is the same for orange and red molecules across the whole temperature range. Moreover, the orbital shift extracted from the intercept is nearly 80 ppm and 220 ppm for orange and red molecules, respectively. These values deviate significantly from the 138 ppm and 143 ppm for orange and red molecules that we extracted
from low temperature measurement for T < Tc, of which further details will be discussed in Section 4.2.2. However, we have to keep in mind that the orbital shift extracted from K − χ analysis is assuming a temperature independent hyperfine coupling constant, whereas the low temperature Knight shift measurement in the superconducting phase directly probes the vanishing spin susceptibility caused by singlet pairing. In short, all of the abovementioned observations indicate a contradiction of the K −χ analysis to other data and call for a further investigation.
To solve this discrepancy, we model the NMR shift K incorporating a varying hyperfine coupling constant A, informed by our T1 ratio analysis. The results are shown in Figure 4.7(c). One expects a more significant change in Knight shift with susceptibility for charge- rich sites. Indeed, if we consider a fixed hyperfine constant A and A , with Ared ∼ 1.4 orange red Aorange extracted from the relaxation rate data, then the charge-rich (red) molecule shows a steeper drop of shift as the temperature decreases (red dashed line). In the case of a temperature de- pendent A, the ratio between Ared and Aorange changes according to temperature dependence evaluated from the T1 data. The expected pronounced drop in shift of the charge-rich site from the susceptibility is now compensated by an increased A on the charge-rich site (caused by charge disproportionation). The modeled NMR shifts in Figure 4.7(c) are obtained by fitting to the following equations:
orange Korange = Aorangeχ + Korb
red Kred = Aredχ + Korb
37 (a) ×10-4 (b) 700
6 600 ) 5 500
mol 4 400 300 3 (emu/mol) (emu/ s
Shift(ppm) 200 Shift (ppm) s χ 2 � 100 1 0 0 0 100 200 300 0246 Temperature (K) χ (emu/mol) -4 Tem per atur e ( K) �s (emu/s mol) ×10 (c) 600 Experiment data 550 Varying HF coupling 500 Fix HF coupling 450 (ppm)
K 400
350
300 Shift (ppm)
250
200 Average Shift Shift Average 150
100 0 50 100 150 200 250 300 TemTemperature per atur e ( (K) K)
Figure 4.7: (a) Macroscopic susceptibility measurement performed by SQUID and ESR [40, 69]; (b) K − χ analysis of orange and red molecules; dashed lines are linearly fit to the data; (c) modeling of NMR shift compared with experimental data; solid/dashed lines are modelings that consider a varying/fixed hyperfine coupling constant A, repectively
orange Ared T1 0.5 f(T ) = = ( red ) Aorange T1
The modeling using T1, together with the susceptibility data agrees well with the observed NMR shift, strengthening the credibility of the relaxation-based charge disproportionation analysis, in the sense that the effect of charge disproportionation can be seen from both hyperfine coupling parallel (Ks) and perpendicular (T1) to the direction of the applied field.
orange red Moreover, the orbital shifts Korb and Korb of approximately 115 ppm and 140 ppm, respec- tively, are closer to the orbital shifts extracted from low temperature measurement comes later. The temperature dependence of ∆K, which is the difference in shift between 13C sites
within the same molecule, remains anomalous in the sense that while ∆Kred drops around
50% from 300 K to T > Tc, T1 ratios within the same molecule are constant upon cooling, as shown in Figure 4.6. This anomaly in ∆K(T ) may be attributed to anisotropic charge
38 dependence of the hyperfine coupling tensor, but further investigation is required. A later analysis in Section 4.3 focusing on the hyperfine shift tensor at T = 300 K shows the ratios of hyperfine coupling constant Ared ∼1.5, validating the T ratio analysis here, at least at Aorange 1 300 K.
4.2.2 Characteristics of superconducting state
Thus far, we have a charge-ordered metal, for which the charge disproportionation increases upon cooling. Next, we examine the superconducting state; NMR is a powerful tool for studying superconductivity by means of dynamic and static measurements. For example, NMR can be used to indirectly probe the superconducting gap structure upon cooling below
Tc. For the case of a conventional s-wave SC, the density of states near EF is modified by the following equation as the gap opened up at Fermi level
E ρsc(E) ∝ √ E2 − ∆2
for E > ∆ + EF, where E is the energy from EF and ∆ is the superconducting gap, whereas −1 for normal state, ρnormal(E) is a constant. For E < ∆ + EF, ρsc(E) = 0. Because T1 scales 2 with ρ(EF) , the divergence of the density of states leads to a peak called the Hebel−Slichter
peak below Tc in the s-wave SC. This phenomenon is predicted in BCS theory and was taken as an important experimental evidence for the BCS framework. Once the superconducting gap ∆ opens up, the NMR spin lattice relaxation is dominated by the thermally excited −∆ −1 k T quasiparticles. For T << Tc, T1 ∝ e B , where kB is the Boltzmann constant. However, for a d-wave SC, the superconducting gap ∆ is not isotropic. For example, for dx2−y2 SC, the gap function is given by
∆(θ) = ∆0(T )cos(2θ)
where θ and ∆0(T ) are the angle in momentum space and superconducting gap magnitude [70]. That is, the d-wave SC has a ”nodal” gap structure, where line nodes with zero magnitude of superconducting gap ∆ exist in the Brillouin zone. With the nodal gap, the divergent density of states is less pronounced compared to the isotropic fully gapped s-
39 −1 wave SCs, and the Hebel–Slichter peak disappears. The temperature dependence of T1 −1 α becomes power law T1 ∝ T rather than exponential dependence, at least in the limit of low temperature.
−1 In Figure 4.8(a), we show the (T1T ) measurement of β”-SC from 1.7 to 80 K where
−1 Tc(B) ∼ 3.4 K at B = 6.87 T [46]. (T1T ) remains constant at higher temperatures, until
the temperature reaches Tc. No evidence for a Hebel–Slichter enhancement is observed below
−1 −1 −3 Tc, and T1 falls off in a power law T1 ∝ T . Notably, the difference in relaxation rate between orange and red ET molecules disappears, indicating a vanishing spin susceptibility, and the relaxation processes are dominated by hyperfine coupling to the quasiparticles near
EF.
(a) (b) 500
10-1
-1 400
300 (K*sec) -2 -1 10 T) Shift (ppm) 1 200 (T
10-3 100 100 101 0246 Temperature (K) Temperature (K)
−1 Figure 4.8: (a) Temperature dependence of (T1T ) for red, orange, and both molecules; (b) temperature dependence of NMR shift K below Tc for red, orange, and both molecules
In Figure 4.8(b), the NMR shift K¯ , which reflects the local spin susceptibility at the
two molecules, is plotted, for temperatures near to and less than Tc. The NMR shift drops
sharply below Tc, showing the formation of a singlet superconducting ground state. For superconducting state with nodal gap structure, a small magnetic field, which causes spin states to split, results in extra quasiparticles near the nodes, unlike in a fully gapped s-wave SC. The temperature dependence of the magnetization of a clean d-wave SC is quadratic for
a large field B >> kBT [71]. Therefore, we fit the NMR shift K¯ with a quadratic function µB of T plus a constant. The extracted shift at T = 0 K for the orange and red molecules are
40 estimated to be 185 and 191 ppm by extrapolation, respectively. Considering the NMR data are collected under a magnetic field of B = 6.87 T at T = 0, approximately 20% of normal state Ks is recovered compared to zero field [71]. Therefore, we estimate the orbital shift
orange red 13 −1 Korb and Korb to be 138 and 143 ppm. Altogether, C NMR T1 and shift measurement on β”-SC indicates that the experimental evidence is consistent with an unconventional SC with singlet pairing and nodal gap.
−1 Particularly notable is the lack of enhancement in (T1T ) above Tc from spin fluctua- tions. This is an important distinction from most other unconventional SCs. The enhance-
−1 ment of (T1T ) , associated to spin fluctuations, occur for T > Tc and is a common property of the normal state in correlated materials. For example, Figure 4.9 contrasts the tempera-
−1 −1 ture dependence of (T1T ) in β”-SC to (TMTSFP)2PF6, a quasi-1D organic SC. (T1T ) is normalized by the value at T = 77 K for each compound. For (TMTSFP)2PF6, a near
−1 fivefold enhancement of (T1T ) compared to high temperature (T = 77 K) is observed,
−1 whereas β”-SC exhibits constant (T1T ) down to Tc. We take this phenomenon together
with increased charge disproportionation approaching Tc as experimental evidence indicating the possibility of a charge fluctuation mediated unconventional SC. (77K) 1 - T 1 T / 1 - T 1 T
Tem per atur e ( K)
−1 Figure 4.9: Comparison of (T1T ) between β”-SC and (TMTSFP)2PF6, a typical SC in −1 proximity to the magnetic ground state. Values are normalized by (T1T ) at T = 77 K
The absence of low-energy antiferromagnetic spin fluctuation, at first glance, seems sim-
ilar to the case of iron-based SCs. That is, LaFeAsO1-xFx compounds show no sign of
41 low-energy antiferromagnetic spin fluctuation above Tc, whereas fluctuations still persist at higher energy, as found in inelastic neutron scattering studies [72–76]. However, unlike
LaFeAsO1-xFx, no long range magnetic ordering in the ground state of this β”-SC SC or any other β” salts with different counter ions was reported [77]. Moreover, in contrast to the
−1 −1 monotonic increase of (T1T ) for T > Tc in LaFeAsO1-xFx,(T1T ) of β”-SC is constant, consistent with typical Fermi liquid behavior for Tc < T < 100 K. Consequently, we argue that β”-SC represents different physics from the case of an iron-based SC.
4.3 Nature of Electronic Correlation: Korringa Ratio
(b) (a) 10 10
5 5
0 Intensity (a.u.) 0 100 200 300 400 500 Shift from TMS (ppm) 0 (kHz)
z 2 a-c plane z f 0 b-c plane -90 a -5 -90 ET molecule c x -10 y x 0 +90 +90 13c y b -10 -5 0 5 10 f (kHz) a 1
c b
Figure 4.10: (a) 13C Top: spectrum obtained at 300 K and B = 8.4530 T with field oriented +92◦ from the a axis in the a-c plane; bottom: rotation relative to crystal and ET molecule orientation; (b) 2D COSY spectrum; (c) and (d) Angular-dependent spectra in shift for a magnetic field oriented in the a-c and b-c planes, where red and orange symbols correspond to the two corresponding molecules
42 To investigate the electron correlation in β”-SC, the Korringa ratio κ is extracted, which is a commonly applied metric for assessing the strength and nature of correlations. For the
2 case of isotropic hyperfine coupling, the product S ≡ Ks T1T reduces to the fundamental constant S0, 2 ~ γe S0 = . 4πkB γn Deviations result from, for instance, a momentum-dependent spin response associated with correlations. For the antiferromagnetic case, the relaxation rate is larger than expected relative to the shift, assuming S0 applies; thus, κ ≡ S0/S > 1 and for the ferromagnetic state, κ < 1. In the case of ET compounds, hyperfine coupling has an anisotropic component associated with the pz characteristics of HOMO. As a result, the Korringa relation must be modified by a geometric form factor [60, 78, 79], β(ζ, η)κ = S0/S, with
(Axx/Azz)2(sin2η + cos2ζcos2η) + (Ayy/Azz)2(cos2η + cos2ζsin2η) + sin2ζ β(ζ, η) = . 2[(Axx/Azz)sin2ζcos2η + (Ayy/Azz)sin2ζ]2
ζ, η are the angles in molecular coordinates, and A = [Axx,Ayy,Azz] is the diagonalized hyperfine coupling tensor with respect to the molecular coordinate, which is commonly taken as the principal axes of hyperfine shift [80], as defined in Figure 4.10. Accordingly, we perform an angular dependent measurement for T1 and shift at room temperature, under rotation in the a-c and b-c planes.
The hyperfine coupling tensor is extracted from the angular-dependent spectra. As dis- cussed in Section 4.2, there are eight peaks in the spectrum that originate from two inequiv- alent 13C sites on each of the two crystallographically distinct molecules. As before, they are labelled as orange and red. In the course of magnetic field rotation, not all sites are resolved in the NMR spectra. Thus, a 2D COSY NMR technique is utilized to provide information of the molecular origin (red vs. orange) of each peak. An output spectrum from such a 2D NMR experiment is shown in Figure 4.10(b), and the associated 1D NMR spectrum is shown in Figure 4.10(a). In the spectrum, dipolar-coupled nuclear spin contributions appear as off-diagonal peaks, and the four peaks that originated from the same molecule are iden- tified, which is sometimes not feasible for the 1D NMR spectrum. Utilizing the 2D NMR
43 technique, the angular-dependent spectra for the two rotation angles are shown in Figure 4.10(c) and (d) with all peaks resolved.
(a) 700 (b) 600
600 500
500 400
400 300 300 200
200 Knight Shift (ppm) Shift Knight Knight Shift (ppm) Shift Knight 100 100
0 0
-100 -100 -80 -60 -40 -20 0 20 40 60 80 100 -80 -60 -40 -20 0 20 40 60 80 Angle in a-c plane (deg) Angle in b-c plane (deg)
(c) 5 (d) 80 a-c plane a-c plane 4.5 b-c plane 60 b-c plane 4
(ms) 40 1 3.5 T 20 3 0 2.5 -100 -50 0 50 100 10 2
1.5 Dipolar splitting (kHz) splitting Dipolar 5 1
0.5 Korringa Ratio Korringa 0 0 -100 -50 0 50 100 -100 -50 0 50 100 Angle in a-c / b-c plane (deg) Angle in a-c / b-c plane (deg)
Figure 4.11: (a) and (b) Angular-dependent Knight shift; solid and dot-dash lines are obtained from the hyperfine tensor; dashed line are the orbital shift for orange and red molecules. (c) Dipolar splitting values plotted vs. rotation angle; the continuous curves cor- respond to the expected variations. (d) Measured T1 at specific orientations (top) Korringa ratios (bottom)
Although the relative rotation angles are recorded accurately by two orthogonal Hall sensors on a single axis rotator, the absolute orientation with respect to crystal axes has a random uncertainty of ±10◦ that is hard to determine given the crystal morphology. Orienta- tion is further constrained by exploiting the angular information contained in the internuclear dipolar split of NMR spectra. This analysis is performed by estimating the magnitude of dipolar splitting D, as given in Equation 4.1, which depends on the angle θ between the applied magnetic field and carbon double bond, the x axis in molecular coordinates. A least-squares fit of the dipolar splitting with respect to angle for both the a-c and b-c planes
44 site Axx (ppm) Ayy (ppm) Azz (ppm) 1 (•) -23 38 413 2 (N) 38 54 625 3 (•) -93 0 624 4 (N) 0 79 984 Table 4.1: Diagonalized components of the hyperfine shift tensor for the four sites. Uncer- tainty is estimated to be ± 20 ppm
xx yy zz Korb (ppm) Korb (ppm) Korb (ppm) Neutral Molecule 66 178 26 Red Molecule (+0.58e) 125 177 55 Orange Molecule (+0.42e) 106 173 56
Table 4.2: Diagonalized components of the orbital shift tensor for ET molecules [81] is shown in Figure 4.11(c) with the initial fitting parameter being the estimated crystal ori- entation, based on crystal morphology. The modeled dipolar splitting agrees well with the experimental data and results in an alignment for both planes to within 5◦. Finally, after obtaining accurate angular information, the orbital shift must be subtracted from the NMR
shift to obtain the Knight shift. Ks(ζ, η) is obtained from
K(ζ, η) = Ks(ζ, η) + Korb(ζ, η)
with Korb evaluated based on the orbital shift tensor reported previously on the α-(BEDT-
TTF)2I3 compound [81].
In Figure 4.11(a) and (b), the angular dependence of the Knight shift for the four sites are plotted. Solid lines are the least-squares fits to the experimental data, using hyperfine
shift tensor Ai of the four sites i, and dashed lines are the orbital shift for orange and red molecules. The Knight shift of the ith site can be written as
" # X `` 2 2 Ki = (Ai H` ) /H0 `
in terms of A and applied field H, where ` =x, y, z. From the fitting, hyperfine shift tensor A is found to agree well with the shift data in both rotation planes. Notably, a good agreement
45 5
4
3
Korringa Ratio 2
1 0 100 200 300 Temperature (K)
Figure 4.12: Temperature dependence of the Korringa ratio of orange and red molecules
of A with shift data in both rotation planes heavily relies on accurate angle determination, which is not possible without modeling dipolar split. Hyperfine shift tensors for the four
5 sites are listed in Table 4.1. For ET molecules, considering onsite dipolar contribution βpz and isotropic Fermi contact contribution α, the hyperfine shift tensor can be written as
xx yy zz A = (A ,A ,A ) = (α − βpz , α − βpz , α + 2βpz ).
Regarding the hyperfine shift tensors, Azz is ∼10–15 times larger than the in-plane compo- nents Axx and Ayy. The anisotropy for the in-plane components that deviates from uniaxial symmetry may be attributed to the neglected off-site pz contributions from neighboring car- bon and sulfur sites, where the off-site dipolar field was estimated to be less than 10% of the onsite dipolar term [80]. Furthermore, the ratio of Azz between red and orange molecules extracted from NMR spectra is consistent with the ratio extracted from NMR relaxation analysis, showing the credibility for both results. The spin lattice relaxation time T1 at
5Offsite dipolar hyperfine field from neighboring carbon and sulfur nuclei is estimated to be 5∼10% of onsite dipolar field and is, therefore, neglected here [80].
46 different orientations of the magnetic field are shown in Figure 4.11(d). We selectively eval- uate the field orientation where spectra are well resolved, so that nuclear spin diffusion does
not play a role. Using T1 data together with shift and Ai at T = 300 K, Korringa ratio
is estimated to be κ '4.1±0.6, consistent with moderate correlations. A temperature de- pendence of the Korringa ratio is evaluated based on the temperature dependent shift K
and T1 in Figure 4.12. For T < 100 K, where β”-SC exhibits approximately constant shift
−1 and (T1T ) , κ is 1.7, indicating correlations are weakened upon cooling. Approaching Tc, κ increases slightly to approximately 2.4. This result agrees with a previous calculation that predicted an antiferromagnetic correlation between spins in a charge-ordered state [15].
For comparison, κ-(BEDT-TTF)2Cu(NCS)2, which shows antiferromagnetic spin fluctuation
approaching the superconducting ground state is estimated to have κ in the range of 7∼9 [60, 82].
4.4 Motion of Ethylene Groups
c
c b
Eclipsed b
a
a
b a
c a c Staggered b
Figure 4.13: Eclipsed and staggered conformations of ET molecules; the view along the carbon double bond (left) and the view perpendicular to the carbon double bond (right)
Ethylene groups at the two terminals of ET molecules are known to have two different conformations, of which the motion affects properties of the organic charge transfer salts. The two conformations are sometimes called staggered and eclipsed, as shown in Figure 4.13. For the staggered state, the two ethylene groups are oriented with opposite twists, whereas the eclipsed state has the ethylene groups aligned, as viewed from the direction of the carbon double bond. The conversion between these two conformations are often ther-
47 mally activated at room temperature, and as temperature decreases either one conformation becomes dominantly favorable or the material undergoes a glassy-like transition where the two conformations of ethylene group coexist. The two conformations have been found to have substantial impact on the electronic system at lower temperatures. For example, for
κ-(BEDT-TTF)2Cu(N(CN)2)Br, a rapid cooling rate, inducing glassiness in molecular orien- tation and possible variation in the electron correlation, is reported to result in an insulating state, whereas a superconducting ground state is observed in the case of slow cooling [83]. In
−1 Section 4.2, we observed an enhanced T1 at T > 100 K that may be thermally activated. Moreover, in a previous high pressure transport study, an insulating state was observed above 10 kbar, which may also relate to ethylene groups [1]. An investigation of the con- formations and motions of ethylene groups is, therefore, required. Here, we study the effect of ethylene groups on β”-SC by NMR of 2H nuclei on the ethylene groups. Notably, our original objective was to use 2H to probe charge fluctuations (2H is a I = 1 nucleus that couples to the electric field gradient). As is shown in the following figure, an enhancement
2 −1 of H(T1T ) induced by charge fluctuation is not observed in β”-SC, possibly due to the small hyperfine field at the terminals of ET molecule. Nevertheless, fruitful information on the motion of ethylene group is revealed.
-4 102 10 ) -1
-6 100 10 (sec 1 (sec) c τ -8 10-2 10 Average 1/T
10-4 10-10 100 150 200 250 300 150 200 250 300 Temperature (K) Temperature (K)
−1 Figure 4.14: Temperature dependence of T1 at B = 8.4525 T; the solid curve is a fitting to the BPP model (left), and (right) shows the temperature dependence of correlation time τc −1 extracted from T1 data
−1 2 In Figure 4.14, the measured spin lattice relaxation rate T1 (T ) of H is shown. A −1 −1 strong temperature dependence of T1 is shown below T = 200 K where T1 drops rapidly.
48 With T1 ∼ 36 s at T = 125 K, collecting NMR data with high signal to noise ratios in a −1 timely manner is impractical below 125 K. T1 (T ) is well-described by relaxation governed −1 by an activated process. The effect from ethylene group on T1 is commonly observed in the 1H NMR of κ phase compounds [80, 84, 85], where the enhanced relaxation rate originates from the motion of nuclei. In the nuclear coordinate, this motion leads to fluctuation of
1 the nuclear dipolar field or electrical field gradient (for I > 2 ), and is maximized when the associated spectral density of fluctuation is largest at the Larmor frequency. We fit T1(T ) with the equations given by the BPP model [86, 87]:
1 τc 4τc = C( 2 2 + 2 2 ) (4.2) T1 1 + ω τc 1 + 4ω τc
∆E k T τc = τ0e B
where τc, τ0, and ∆E are the correlation time, correlation time prefactor, and the activation energy of the motion, respectively; and ω and C are the Larmor frequency and a constant related to the square of fluctuation magnitude, respectively. The fitting parameters obtained
−13 10 −2 ∆E are τ0 = 4.3 × 10 s, C = 1.3 × 10 s and ∼ 2000 K. These fitting parameters kB describe the rapid reduction below T = 200 K fairly well. To confirm that this temperature
−1 −1 dependence in T1 is a dynamical effect, we measure T1 at three different fields. The −1 maximum T1 is expected to occur at higher temperatures for higher frequencies. Indeed, −1 shifts of T1 (T ) are observed in the experiment and the modeling works well, as shown in −1 Figure 4.15. We model the T1 data at different fields with a single set of fitting parameters
τ0, C, and ∆E, and only change ω based on the Larmor frequency at the given field.
Here, the following questions arise: (1) What is the origin of the fluctuation field? (2) What is this motion of ethylene group’s effect on the 13C relaxation as discussed in Section 4.2? We perform an estimation of the order of magnitude for the constant C to clarify the origin of fluctuation; for spin lattice relaxation assisted by nuclear dipolar coupling and electric quadrupolar coupling, the constant C is given by the following equations [88, 89].
2 2 Cdip ∼ 0.3Bdip(γ2H)
49 9.3512T 2 6.3342T 10 2.8611T ) ) -1 -1 (sec (sec 1 1 0 150 T 10 1/T ) 1/ -1 100 (sec 1 T
1/ 50
0 100 150 200 250 300 10-2 Temperature (K) 100 150 200 250 300 TemperatureTemperature (K) (K)
2 −1 Figure 4.15: Temperature dependence of H T1 at different fields; solid lines are fittings to the BPP model at three different fields, and thus three different Larmor frequencies. The inset are the same data but plotted in a linear y axis
1 3eV Q C ∼ ( zz )2 quadrupolar 3 4h
with ~γ2HI Bdip = 3 (r2H−2H) 2q Vzz = 3 (r12C−2H)
γ2 2H MHz where 2π = 6.53593 T , e = one electron charge, Q2H = 0.0027 barn is the quadrupole moment of 2H, h is Planck’s constant, I=1 is the nuclei angular momentum for 2H and
2 r2H−2H ∼ 1.55 A˚ is the distance between nearby H nuclei. In our estimation, we consider the dipolar field from neighboring 2H nuclei. For quadrupolar coupling, we take q = +1e on
12 7 −2 C. Altogether, the estimation gives Cdip ∼ 2 × 10 s with Bdip ∼ 0.6 G and Cquadrupolar ∼
9 −2 10 7.5×10 s . Cquadrupolar is much closer to our experimental fitting parameter C = 1.3×10 s−2, and consequently, we infer that the relaxation of 2H NMR is dominated by quadrupolar coupling.
50 0.2 ) -1 6.87T 10-1 8.4530T sec -1 ) ) (K -1 0.15 T 1
T 10-2 1/( sec 102 -1 Temperature (K) 0.1 T (K 1
1/T 0.05
0 0 100 200 300 Temperature (K)
13 −1 Figure 4.16: Revisiting temperature dependence of C NMR (T1T ) for red and orange −1 molecules; the solid line is (T1T ) of orange molecules plus a temperature independent −1 constant. This figure shows the hyperfine origin of (T1T ) enhancement above T = 100 K. −1 The inset is a comparison of the mean (T1T ) of both molecules measured at B = 6.87 T and 8.4530 T
Furthermore, the enhanced relaxation of 13C at T > 100 K shown in Section 4.2 should
13 1 be addressed. First, C is a spin I = 2 nucleus that does not couple to an electrical field gradient. Second, considering that the distance from the ethylene group to the carbon double bond is approximately 5 A,˚ the dipolar field from 1H on the ethylene group is of the order of 0.1 G, which is too small to account for the enhancement. Moreover, if the enhanced relaxation of 13C originates from the 1H dipolar field, it should contribute equally to both molecules if we consider ethylene groups on both sides of ET molecules. In Figure 4.16,
−1 we take (T1T ) of the orange molecule and add a constant. Thus, the low temperature
−1 −1 (T1T ) matches the low temperature (T1T ) of the red molecule. Then, it is obvious that
−1 the enhancement of (T1T ) above 100 K does not increase uniformly with temperature, but strengthens with larger hyperfine coupling. The motion of ethylene group can cause modulation of the HOMO orbital energy as proposed in [84], leading to fluctuation of the hole density and, therefore, enhanced relaxation at the 13C site. However, if this is the
51 −1 dominant effect of ethylene motion, T1 above 100 K will be strongly field dependent. The −1 inset of Figure 4.16 compares (T1T ) measured at B = 6.87 T and 8.4530 T. Considering
−1 −1 1 the effect of ethylene motion to be in a slow limit, where τ >> ω , then T ∝ 2 . At c 1 ω τc −1 −1 B =6.87 T, (T1T ) should be 1.5 times higher then (T1T ) at B = 8.4530 T, which is not what we observed. Another effect of ethylene motion can be the change of electron correlation at T > 100 K as reflected in Figure 4.12. This can be the case if, for example, intermolecule distance is changed by the ethylene dynamics. At T >100 K, the Korringa ratio κ increases monotonically and reaches ∼ 4.6 at 300 K. As electron correlation strengths, charge disproportionation is suppressed. Whether the ethylene motion corresponds to the charge disproportionation in β”-SC is, however, uncertain at this point. The monotonic temperature dependence of charge disproportionation does not seem to be affected strongly at T ∼ 100 K where the motion ceases. At the same time, the uncertainties in the analysis of charge disproportionation may not give us sufficient precision to exclude a small change in slope at T ∼ 100 K.
The effect of ethylene motion on 13C NMR has been reported before on a dimerized κ, β
compound, although only in T2 [84, 90, 91]. For our case, we do not see any peak broadening in 13C NMR, at least for T > 100 K, at which temperature the motion of ethylene groups
is still active. We attribute this phenomenon to the short correlation time τc in β”-SC,
−1 −1 −1 as well as a higher baseline of (T1T ) ∼ 0.1 K sec in other ET compounds caused by
antiferromagnetic spin fluctuation [60]. In β”-SC, τc is approximately 1 µs << T2 ( 100 µs)
at T ∼ 100 K as shown in Figure 4.14 (right). For τc << T2, nuclei exhibit a motionally
narrowed peak. Only for τc values similar to or greater than T2, does the NMR spectrum −1 become sensitive to the fluctuation field, resulting in an enhanced T2 and homogeneously
broadened spectrum (τc ∼ T2), or an inhomogeneously broadened spectrum (τc >> T2). For
−1 the quarter-filled compounds, a sharp (T1T ) enhancement is also observed in θ compound and is attributed to a crossover from Fermi liquid to bad metal due to charge fluctuation [79]. Here, based on the 2H and 13C NMR measurement, we suggest this relaxation enhancement
13 −1 2 −1 in C(T1T ) , which occurs in the same temperature range as enhanced H T1 , may be due to the motion of ethylene groups that causes increased correlation.
52 Figure 4.17: Temperature dependence of 2H NMR spectra measured at B=8.4525 T; the x axis plots deviation from zero field frequency ω0 at this field f=55.247 MHz; intensity of spectra at various temperatures are normalized
Next, to understand how the motion of the two conformations of ethylene groups in- fluences the NMR spectra, the temperature dependence of the 2H spectrum is studied in Figure 4.17. In total, 16 different 2H sites exist for β”-SC; with quadrupolar splitting of the I = 1 nucleus, it results in 32 peaks. The population of conduction electrons at the two ends of ET molecules is nearly zero based on the abovementioned calculation. As a result, the spectra for the whole temperature range exhibit negligible NMR shifts and sit near the
zero field frequency ω0 = 55.247 MHz at B=8.4525 T. The quadrupolar splitting of differ- ent peaks increase or decrease as temperature drops and no significant broadening in NMR
peaks is observed. This phenomenon can be interpreted using the same correlation time τc
2 extracted from H T1 measurement and the crystal information of the two conformations
2 of ethylene groups. Here, τc is in the fast limit, where τc << T2. Therefore, H nuclear spins experience a motionally-averaged electric field gradient associated with staggered and eclipsed conformation, weighted by the conformation ratio. The ground-state conformation
53 is related to the Van der Waals coupling between the ethylene groups and anion layers. For β”-SC, a structural study showed that at T = 123 K, most of the ET molecules are in eclipsed form [37]. Accordingly, we simulate the 2H spectrum in Figure 4.18 based on
ET molecules in eclipsed form the reported crystal structure. The y axis is the order ratio = all ET molecules . The simulation qualitatively verifies that the temperature dependence of increasing or decreasing quadrupolar splitting is not due to an increased or decreased local electric field gradient, but may instead originate from a change in the ratio between two conformations at different temperature. The discrepancy may creep in from the simplified point charge distribution, which we assumed in addition to the missing information of exact structure of staggered conformation that has not been previously reported. In the simulation, we merely take the position of mirror reflection of the ethylene group with respect to the molecular plane, de- fined by the S-C-S bond at the central carbon double bond, to be the nuclear coordinate, in the staggered state.
0.4
0.5
0.6
0.7
0.8 Eclipsed Ratio
0.9
1
-150 -100 -50 0 50 100 150 Frequency (kHz)
Figure 4.18: Simulation of 2H NMR spectrum based on quadrupolar split of 2H caused by +2e on carbon sites at the fast limit of τc << T2
54 4.5 Response Under Pressure
In [15], the calculated phase diagram of a quarter-filled system is tuned via V/t, which presents the strength of electron correlation. Similarly, we study the effect of electron corre- lation tuned via pressure in β”-SC. We measure 13C NMR in a BeCu pressure cell at three
different pressures P = 0.25, 5, and 8 kbar calibrated at room temperature by CuO2 [92]. The measurement is performed at B = 8.4525 T and T = 1.5 ∼ 300 K. In Figure 4.19,
−1 we show the temperature dependence of (T1T ) at different pressures. For P = 0.25 kbar,
−1 (T1T ) reproduces our previous data because the pressure is low. For P = 5 and 8 kbar,
−1 (T1T ) remains constant below T < 100 K, showing metallic behavior that is the same as
−1 ambient pressure. For T > 100 K, the enhancement in (T1T ) is suppressed with higher pressure.
0.25 kbar 5 kbar ) ) -1 -1 8 kbar -1 10 K -1 sec -1 ) (K T (sec T 1 1 T 1/T 1/( 10-2
101 102 TemTemperatureperature ( K(K))
−1 Figure 4.19: (T1T ) at various pressures are plotted as a function of temperature; solid lines are fittings to Equation 4.2
−1 First, we examine the value of (T1T ) in the metallic region for Tc < T < 100 K. As pressure increases and the distance r between sites becomes smaller, the transfer integral t in the aforementioned Extended Hubbard model increases exponentially. Therefore, electron
55 correlation that presents in U/t or V /t decreases, where U and V are onsite and intersite Coulomb repulsion, respectively, as discussed in Equation 1.2. Increased t results in smaller
−1 density of states N(E). Hence, (T1T ) in the metallic region, which can be written as
−1 2 −1 (T1T ) ∝ N(EF) , decreases. From 0.25 to 8 kbar, (T1T ) drops monotonically from
−1 −1 0.0207 to 0.0106 sec K giving a suppression of N(EF) approximately 5.0%/kbar, similar to results that have previously been reported for κ compounds [93, 94]. This value also agrees with the change in effective mass under pressure in transport measurement [1].
0.25 kbar 3 5 kbar 8 kbar
0.5 2.5 Ratio) 1 2 (T
1.5
0 100 200 300 Temperature (K)
Figure 4.20: Temperature and pressure dependence of charge disproportionation are ex- tracted from T1 data of the lowest and highest frequency sites, which correspond to redout and orangein. Solid lines are linear for T > 50 K
The response of charge disproportionation under pressure can be studied using a similar approach to in Section 4.2, although we looked at the T1 ratio between the lowest and highest frequency sites instead of orange and red molecules to avoid overlapping peaks; the result are shown in Figure 4.20. When P =0.25 kbar, charge disproportionation increases monotoni- cally, reproducing our analysis for orange and red molecules at ambient pressure. As pressure increases, charge disproportionation is suppressed as the temperature dependence becomes weaker, as shown by the linear fit. This phenomenon is expected since electron correlation
56 becomes weaker with increased t and, therefore, suppressed V/t at higher pressures. Similar observations have been reported in another quarter-filled system α compound, which is also known to exhibit charge disproportionation [95, 96]. In addition, a distinct finding is the sudden drop of charge disproportionation at T < 50 K, which enhances with pressure. This rapid change of charge disproportionation may be related to a possible band reconstruction suggested by transport measurement in β”-SC at high pressures [1]. The same report also revealed a first-order metal-insulator transition at P = 9-12 kbar, which would be interesting to investigate using NMR, and is an on-going experimental plan at this moment.
In summary, we report an example of unconventional SCs, of which the superconduc- tivity may be mediated by charge fluctuation, supported by the absence of low-energy spin fluctuations, a simultaneous suppression of superconductivity and charge ordering by pres- sure, and a weak to moderate electron correlation of Korringa ratio κ ∼ 2.4 approaching
Tc. Moreover, the ethylene motion in β”-SC is found to play a role in modifying electron correlation, which can be a general phenomenon for organic charge transfer salts. Our ob- servations demonstrate the novel properties of the quarter-filled q2D organic SC β”-SC, and certainly call for further studies.
57 CHAPTER 5
Result II: Properties of the FFLO States
Since the prediction of FFLO state in the mid 1960s, there was little experimental evidence for its stabilization, especially microscopic evidence [21, 22]. Reports included heat capacity, magnetic torque, penetration depth, etc. are informative for studying the enhanced upper
critical field Hc2, or specifically, the upturn of Hc2 in the H-T phase diagram of a Pauli-limited SC, accompanying the first-order phase transition from uniform superconducting state (uSC) to FFLO state [46, 53, 97]. However, microscopic evidence, such as a direct probe of the order parameter modulation, is difficult to obtain, and thus far, no reports from scanning tunneling microscopy or neutron scattering experiments exist. Nevertheless, some evidence for the FFLO state from NMR investigations were reported as discussed in Chapter 2. An NMR study on β”-SC provided spectroscopic evidence consistent with incommensurate 1D modulation of superconducting order parameter [55]. Another microscopic NMR study on
κ-(BEDT-TTF)2Cu(NCS)2 reported an enhancement of NMR spin lattice relaxation rate in the FFLO state, which was attributed to the nodes in the order parameter modulation, the so-called Andreev Bound State (ABS) [36]. The latter phenomenon is the key topic under study in Chapter 5. In Section 5.1, we examine the NMR spectra collected at various magnetic fields and discuss the spectra of the uSC state, FFLO state and normal state in
−1 β”-SC. In Section 5.2, we examine the enhancement of (T1T ) in the FFLO state in β”-
−1 SC for in-plane and out-of-plane magnetic fields. We observe an enhancement in (T1T )
−1 similar to that reported in [36]. Moreover, we find a giant enhancement of (T1T ) in the
−1 FFLO state that is 2.5 times faster than in the normal state (T1T ) when the magnetic field is misaligned by ∼ 1◦ respect to the conducting plane. In Section 5.3, we discuss the experimental results in the context of the underlying physics of the in-plane and out-of-plane
58 −1 (T1T ) enhancement, and suggest that this enhancement, for both in-plane and out-of-plane magnetic fields, are of the same origin and can be attributed to the motion of nodes in the modulation of order parameters in a fluctuating FFLO state.
5.1 Spectroscopic Study of the FFLO state
To study the properties of the FFLO state that exists below T = 1.7 K in β”-SC, we conduct our NMR measurement using top tuning in two different setups that are capable of reaching temperatures below 1 K. One is a dilution refrigerator, configured with a 12 T superconducting magnet at UCLA. This system is mainly used for field sweeps, and the other is a single shot 3He close cycle system of holding time ∼40 hours, equipped with a 16 T magnet, located at Dresden High Magnetic Field Laboratory, Germany (HLD).1 In both setups, the orientation of magnetic fields is aligned to the a-b plane with an uncertainty of ±0.05◦ and perpendicular to the b axis with an uncertainty of 5◦∼10◦, using the methods discussed in Section 4.1 with either a mechanical or piezo rotator. Reported fields and temperatures are calibrated carefully before and after each measurement using the Cu NMR signal from the coil.
The 13C NMR spectra for different in-plane magnetic field strengths B = 6 ∼ 12 T, recorded at T = 749 mK, are shown in Figure 5.1. The parameter space in the phase diagram is presented by the red solid arrow in Figure 5.2. In the right-hand panel of Figure
5.1, the first moment M1 and second moment M2 are plotted, of which the mathematical expressions are given here.
R I(ω)ωdω First Moment M = 1 R I(ω)dω
R I(ω)ω2dω R I(ω)ωdω Second Moment M = − [ ]2, 2 R I(ω)dω R I(ω)dω
1With limiting field and temperature dynamic range in our lab at UCLA, we collaborate closely with S. Molatta, H. K¨uhne and J. Wosnitza from HLD, Germany. Part of the data in this chapter are obtained mainly by S. Molatta.
59 13 300
280 Tonset 260 12 240 (ppm)
1 220
11 M 200
First Moment (ppm) 180 10 160 6 7 8 9 10 11 12 FieldField (T)(T) 9 Field (T)
Field (T) 90
8 80 (ppm) 0.5 70 (ppm)
7 0.5 ) 2 60 M (
50 6 Second Moment 40 0 100 200 300 400 500 600 6 7 8 9 10 11 12 ShiftShift (ppm) (ppm) FieldField (T)(T)
Figure 5.1: Field dependence of 13C NMR spectra at T = 0.749 K (left); first moment analysis (upper right); and second moment analysis (lower right) where ω and I(ω) are the angular frequency, and the intensity at angular frequency ω, respectively. The first moment in NMR spectra presents the average shift, or equivalent spin polarization, weighted by intensity. For equivalent sites, the second moment characterizes the width of NMR peaks. However, for spectra composed of inequivalent sites, as in the case of β”-SC, the second moment has mixed information of the average shift and width and is rather difficult to interpret, especially when the difference in shift between sites is large. We first look at the spectrum in the uSC state, for instance, at B = 6 T, where uSC state refers to uniform superconducting state. In uSC state with singlet pairing, the hyperfine shift
Ks vanishes in the limit of low magnetic field, and only the orbital shift Korb remains for
T << Tc. For β”-SC, the difference in Korb between sites is small and of the order of 10 ppm
(Chapter 4). Therefore, at T << Tc in the uSC state, NMR peaks of the four inequivalent sites overlap, and the spectrum only shows one broad peak as shown in 6 T spectrum in the
60 figure.
For a nodal superconducting gap under the magnetic field, the quasiparticle spin degener- acy is lifted and the spin susceptibility increases linearly with field. As a result, the observed NMR shift increases linearly with applied field between B = 6 ∼9 T. At B ∼ 9.25 T, the first moment increases discontinuously due to an increased spin polarization at the nodes in the modulation of order parameters in the FFLO state, and signifies the phase transition to FFLO state. While the transition from uSC to FFLO state is a first-order transition as shown by calorimetric measurements in [53], the shift may be broadened by disorder. Above
dM1(B) B ∼ 10 T, the slope of the fiend dependence of the first moment dB becomes more gentle. Nevertheless, superconductivity remains up to B = 12 T, as indicated by the increasing first moment of NMR spectra with temperature at B = 12 T as previously reported in [55]. The increase of the second moment at low field (B = 6 T) may be attributed to nuclear dipolar broadening from neighboring nuclei, which is independent of field, and consequently becomes significant in shift at a lower magnetic field. In sum, the transition field from uSC to FFLO state is found to be at 9.25 T at 749 mK.
14 Normal 12 FFLO
10
Field (T) 8 Field (T)
6 u-SC
4 01234 Temperature (K)(K)
Figure 5.2: Plot of temperature and field sweep for an in-plane magnetic field; solid/dashed arrows correspond to the field/temperature sweep shown in Figure 5.3[46, 98]
61 −1 5.2 Enhancement of (T1T ) in the FFLO state
5.2.1 Enhancement under in-plane magnetic field
Next, the NMR spin lattice relaxation rate in the FFLO state is examined to study ex- citations originated from the modulation of order parameters. In Figure 5.2, we plot the experimental parameter space of temperature and field sweep shown by dashed and solid arrows in the phase diagram. The solid line in the phase diagram was the calculated phase boundary reported in [98], where T > T *∼ 1.7 K is the FFLO–normal state phase bound- ary and T < T * is the uSC–normal state phase boundary. Temperature and field depen-
−1 dences of (T1T ) are plotted in the left and right panels of Figure 5.3. At T = 323 mK,
−1 −1 (T1T ) increases gradually with field, approaching the normal state value (T1T ) ∼ 0.019
−1 −1 −1 sec K , and at all fields, (T1T ) is suppressed below normal state value, as a consequence of the singlet paring of superconductivity. As the temperature T increases, the enhanced
−1 (T1T ) (above normal state) is maximized around B ∼10 T at T = 749 mK, reaching an astonishingly large value, namely 30% higher than the normal state. For comparison, for
−1 T = 2 K > T * in the inset, where the transition is from uSC to normal state, (T1T ) drops relative to the normal state value for H < Hc2 ∼ 9 T, and no enhancement is observed. In the temperature sweep in the right panel of Figure 5.3, we again observe an enhancement
−1 −1 of (T1T ) ∼ 30% above normal state (T1T ) at B = 10 T, which is rapidly suppressed at
−1 higher fields. To summarize qualitatively, this (T1T ) enhancement exists only in the FFLO state and is maximized at the applied magnetic field near FFLO–uSC phase boundary and at higher temperatures.
−1 As stated, the enhancement of (T1T ) above normal state is unexpected, considering the
−1 first moment M1, and (T1T ) , shown in the field sweep in Figure 5.4. M1 of the spectrum at B ∼10 T is below the normal state value of 283 ppm, showing the loss of spin susceptibility
−1 −1 due to singlet superconducting pairing; (T1T ) at 10 T goes above normal (T1T ) , indi- cating physics other than thermally excited quasiparticles near the Fermi surface. A similar
−1 enhancement of (T1T ) was previously reported in κ-(BEDT-TTF)2Cu(NCS)2 [36]. Our result shows that the effect also clearly exists in β”-SC and may be a general phenomenon
62 0.03 0.025 322 mK 10 T 533 mK 10.5 T ) ) ) ) 11.5 T -1 0.02 749 mK -1 0.025 -1 -1
(T1T)-1normal sec sec sec 0.015 sec (T1T)-1normal -1 -1 -1 -1 0.02 (K (K
) (K 0.025 ) (K T T
T 0.01 T 1 1 )
1 0.02 1
-1 2 K T T sec T 0.015 T -1 (K
1/ 1/ 0.015 T 0.01 1 T 1/( 1/(
0.005 1/ 0.005
0 6 8 10 12 Field (T) 0 0.01 6 8 10 12 100 Field (T) Temperature (K)(K)
−1 Figure 5.3: Magnetic field (right) and temperature (right) dependence of (T1T ) enhance- ment in the FFLO state under in-plane magnetic field. The inset in the left panel shows the field sweep data at T = 2 K in contrast to data at T = 0.749 K with the x, y scale being the same as in the main figure for the FFLO state. A key issue is then to identify the origin and the relationship to the FFLO state, and will be the main focus of the rest of this Chapter. Regarding the phase
dM2(B) transition from FFLO to normal state, in [36], the change of dB was assigned as the
upper critical field Hc2 from an FFLO to normal state for κ-(BEDT-TTF)2Cu(NCS)2. In β”-SC at B = 10 T, the NMR shift is still below normal state value, and we find the field dependence of M1 to be rather weak, and therefore, it is difficult to identify Hc2 precisely using first moment analysis.
350 First1st moment Moment M1 0.025 (1/T1T)-1T at 749749mK mK 1 1/( ) T
0.02 -1 1
300 T ) (K normal sec Ks 0.015 -1 -1 (ppm) 1 (K sec T
M 250 0.01 1 T -1 1/ )
First Moment (ppm) 0.005 200 0 6 8 10 12 Field (T)
−1 Figure 5.4: Comparison of the field dependence in (T1T ) and the first moment; normal state shift is approximately 283 ppm obtained from the spectrum at B = 12 T, and T = 1.9 K and is indicated by the dashed line
63 5.2.2 Enhancement under out-of-plane Magnetic Field
2
1.5
0.07 1 1stFirst moment Moment M1 280 1/T T-1 at 790mK (T1T1) at 790 mK 0.06 1/( ) T -1 1
270 0.05 T
0.5 ) (K sec plane (deg) -1 plane (deg) b K -1 b 0.04 - ( - (ppm)
sec a
1 260 a T 1 M 0 0.03 T -1 1/ )
First Moment (ppm) 250 0.02 (T1T)-1normal Angle from 240 0.01
Angle from Angle from -0.5 -2 -1 0 1 2 3 AngleAngle from from a a-b-b plane plane (deg) (deg)
-1
-1.5 0 100 200 300 400 500 ShiftShift (ppm) (ppm)
Figure 5.5: Angular dependence of spectra in the FFLO state at B = 9.25 T and T = 0.79 −1 K (left) and comparison of (T1T ) and the first moment (right)
To further understand the underlying physics of NMR relaxation in the FFLO state,
−1 the angular dependence of (T1T ) close to the uSC–normal phase boundary specifically at B = 9.25 T is investigated as shown in Figure 5.5; this was originally recorded for the purpose of field alignment. Starting by looking at the angular dependent spectra, we first notice how sensitive the spectrum is to a misaligned field. As the magnetic field is
◦ ◦ tilted, M1 grows sharply from 0 ∼ 1.5 , indicating increasing density of quasiparticles.
−1 Simultaneously, the angular dependence of (T1T ) turns out to be rather surprising. We find
−1 a huge increment of (T1T ) , increasing linearly with angle, measured relative to the aligned condition, and reaches its maximum, 0.55 sec−1K−1, at 1◦ misaligned angle, which is nearly
−1 2.5 times the normal state (T1T ) . One would intuitively consider the contribution from Abrikosov vortices when the field is misaligned. Our analysis shows that this enhancement
64 cannot be a result of current from vortices, which we discuss in detail in Section 5.3. Indeed, if an orbital shift of ∼ 150 ppm is assumed (Chapter 4), the change in Knight shift Ks is
◦ −1 approximately 35% at 1 misalignment. According to Korringa law, an increment in (T1T ) corresponds to the enhanced spin susceptibility that can be estimated to be approximately
−1 80% compared with the aligned condition. In this case, (T1T ) increases by nearly 250%.
0.06 0.05 10T T=790 mK B=10 T 0.4 K 0.05 9.25T 0.7 K ) ) ) 8.25T ) 0.04 -1 -1
-1 -1 1.05 K 0.04 6.5T sec sec sec sec -1 -1 -1 -1 0.03 0.03 (K (K
) (K (T1T)-1normal ) (K T T T T 1 1 1 0.02 1 0.02 T T T T
1/ 1/ -1 1/( 0.01 1/( (T1T) normal 0.01 0 -2 -1 0 1 2 -2 -1 0 1 2 3 Angle fromfrom a a--bb plane plane (deg) (deg) Angle fromAngle a -(deg)b plane (deg)
−1 Figure 5.6: Magnetic field (left) and temperature (right) dependence of (T1T ) enhancement in the FFLO state under out-of-plane magnetic fields
−1 In Figure 5.6, the properties of this out-of-plane enhancement of (T1T ) are further investigated by probing its response at various temperatures and fields. First, at T = 790
−1 mK, for B = 8.25 T and below, (T1T ) increases toward the normal state value gradually as the field is tilted. At B = 9.25 T, the onset field of the sharp first moment increment
−1 under the in-plane field shown in Section 5.1, (T1T ) exhibits the aforementioned huge enhancement. At a slightly higher field of B = 10 T, the out-of-plane enhancement is suppressed and the maximum enhancement of approximately 1.5 times the normal state
−1 ◦ ◦ −1 (T1T ) occurs at ∼0.4 . For a misaligned angle >1 ,(T1T ) decreases gradually toward
−1 the normal state value. At B = 10 T, we collect the angular dependence of (T1T ) at
−1 three different temperatures. Unlike the enhanced (T1T ) at higher temperatures for a
◦ −1 field aligned at 0 , this out-of-plane (T1T ) enhancement dies out at higher temperatures. From T = 0.4 to 1.05 K, the tilted angle where the maximum enhancement occurs decreases from 1◦ to approximately 0.4◦. The following sections discuss whether this enhancement is intrinsic of an electronic spin environment, and if so, whether it is governed by the same
65 −1 physics that cause the in-plane enhancement of (T1T ) .
−1 5.3 Origin of Enhancement of (T1T ) and the Nature of the FFLO state
5.3.1 Mechanism of NMR spin lattice relaxation revealed by a molecule-specific
T1 study
A critical question to ask at this point is what is the origin of the out-of-plane and in-
−1 plane enhancement of (T1T ) . Does it theoretically agree with what is expected for the FFLO state? For example, can it come from magnetic flux penetrate through as vortices? For a magnetic field aligned in-plane, vortices penetrate through the insulating layers and are coreless. Therefore, no normal state quasiparticle is created, which will contribute to relaxation. For a field aligned out-of-plane, a limited amount of Abrikosov vortices can exist. Nevertheless, the relaxation assisted by the vortices’ cores should not exceed the normal state value. In addition to partially destroying superconducting pairing, vortices facilitate NMR spin lattice relaxation through a fluctuation field produced by vortex motion. In this scenario, the fluctuation field induced from vortex motion should be the same on the charge-rich (red) and -poor (orange) sites. Accordingly, if the T1 ratio between the charge-rich and -poor molecules is close to one, then the enhancement may come from vortices, whereas if the
−1 2 main relaxation mechanism is through hyperfine coupling to the pz electrons, then T1 ∝ A , where A is the hyperfine coupling constant and is proportional to the hole density on the molecules [65]. Here, we study the T1 ratio between charge rich and poor molecules for the
−1 out-of-plane and in-plane condition where the (T1T ) enhancement is strong; specifically, the out-of-plane enhancement at B = 9.25T and the in-plane enhancement at B = 10.5 T.
In general, the spectra in an SC state is narrower due to a smaller hyperfine shift; thus, unlike normal state, peaks from inequivalent sites are not resolved and are difficult to analyze separately. Consequently, we attempt to disentangle the T1 of red and orange molecules by integrating over appropriate spectral ranges. Specifically, we integrate the spectrum in the
66 3
10-1 ) -1 ) -1
K 2.5 sec -1 -1 ) (K T (sec T 1 1
T 2 1/T 1/(
10-2
plane (deg) 1.5 0123b - AnAnglegle from from a-b a -pblane plane (de g(deg)ree) a
4.5 1 4 1 Angle from a-b plane (deg) 1 Angle from Angle from T 3.5 0.5 3 Ratio of T Ratio of 2.5
2 0
1.5 0123 01234 AngleAngle from alignedfrom a -conditionb plane (deg) (degree) FrequencyFrequency (Hz) ×104
−1 Figure 5.7: Angular dependence of (T1T ) of orange and red molecules at B = 9.25 T, T = 0.790 K (upper left); T1 ratio between orange and red molecules at different angles (lower left) angular dependence of spectra. Orange and red regions indicate the integration range used to evaluate molecule specific T1 (right)
normalized spectral weights of 25 ± 20% and 75 ± 20%, as shown in the right of Figure
−1 5.7;(T1T ) is plotted on the left. The angular dependence of orange and red molecules
−1 basically follows the (T1T ) obtained from the whole spectrum, as demonstrated in Figure
5.6. The ratio between red and orange T1 increases from approximately 2 to 4.2 near to
◦ 1 misalignment, approaching the T1 ratio ∼4.4 extracted in Section 4.2, and drops slightly after. This is clear evidence that the enhancement at 1◦ of nearly 2.5 times the normal state
−1 (T1T ) originates from hyperfine coupling instead of the macroscopic field from vortices. A lower ratio below 1◦ does not necessarily indicate that the in-plane enhancement is not of hyperfine origin, but does reflect the fact that nuclear spin diffusion comes into play for overlapped peaks in the spectra. Indeed, when the temperature sweep at B = 10.5 T is examined in Figure 5.8, T1 ratio between red and orange molecule reaches ∼4.6 at T = 1.2 K,
67 −1 the temperature at which the peak of (T1T ) enhancement occurs. Evidently, the in-plane enhancement also originates from hyperfine coupling, and we discuss the possible origin of enhancement based on these analyses in the following section.
5 ) )
) 4.5 -1 -1 -1 1 1 T sec sec sec 4
-1 -1 0.024 -1
0.022 ) ) -1 -1
(K 0.02 ) (K ) (K sec sec -1 T 3.5 -1
T T 0.018 1 ) (K (K 1 1 T T 1 Ratio of T 1 Ratio of T
T 0.016 T T T 1/( 1/ 0.014 1/
1/( 1/( 3 0.012 -2 100 10 TemperatureTemperature (K) 2.5 0.4 0.6 0.8 1 2 3 4 5 0.4 0.6 0.8 1 2 3 4 5 TemperatureTemperature (K) (K) TemperatureTemperature (K) (K)
−1 Figure 5.8: Temperature dependence of (T1T ) of orange and red molecules under an in-plane field at B = 10.5 T (left) with T1 ratio between orange and red molecule; the −1 dashed line indicates the temperature at which (T1T ) enhancement is maximized (right)
5.3.2 Discussion on the possible origins of relaxation enhancement
−1 The previously reported in-plane enhancement of (T1T ) in the candidate FFLO material
κ-(BEDT-TTF)2Cu(NCS)2 was interpreted as originating from the localized density of states at the node of order parameter modulation, namely the ABS [36]. Here, we make an order
−1 of magnitude estimation of the effect of ABS on (T1T ) , and show that other effects may need to be considered to explain the enhancement in NMR relaxation. The structure of density of states is described by a d-wave gap function and ABSs are considered as a delta function at where superconducting gap ∆ ∼ 0 in the momentum space, as shown in Figure
−1 5.9, following the description in [71]. T1 is given by
1 Z ∝ N(E)N(E + ~ω)f(E)(1 − f(E + ~ω))dE, T1 where ~ω is the nuclear Zeeman splitting and can be omitted because it is small (∼5 mK at B = 9.3 T); f is the Fermi-Dirac distribution function and N(E) is the density of states. In
68 a metallic phase,
1 normal 2 ( ) ∝ N(EF) kBT . T1
1D kF Assuming a simplified 1D case, the density of states can be written as N (EF) = , normal 4πEF −1 and the contribution of ABS on T1 is
Z 1 ABS ( ) ∝ NnormalNABSf(E)(1 − f(E))dE T1
δ(E − E ) N 1D = z , ABS λ
where λ is the period of modulation and Ez is the electronic Zeeman energy, respectively. We estimate λ to be at the order of coherence length. Consequently,
1 1 π∆ πµ B k ∼ = = √B p F , λ ξ ~vF 2EF where µB and BP are the Bohr magneton and Pauli-limiting field, respectively. The integra- tion of delta function gives
1 ABS 1 kF 1 ( ) ∝ Nnormal f(Ez) = f(Ez), T1 λ 4πEF λ
−1 where (1 − f(Ez)) ∼ 1. Finally, we look at the contribution of ABS on (T1T ) normalized
−1 by a normal state (T1T ) : √ (T −1)ABS 2π2E 1 = z f(E ). −1 normal z (T1 ) kBT
Ez −5 For T = 0.75 K, B = 10.5 T, is ∼ 9.4 and f(Ez) ∼ 8 × 10 , kBT
(T −1)ABS 1 ∼ 1%. −1 normal (T1 )
Therefore, it is unlikely that ABS can provide an enhancement of approximately 35% of
−1 normal state (T1T ) under an in-plane field alignment. Moreover, as shown in Section 5.3.1, the enhancement scales with hyperfine coupling, and hence cannot be the effect of
69 vortices. For supporting evidence, we also attempt a 19F NMR measurement. With nearly
−1 zero spin population density near the Fermi surface, the hyperfine contribution to its (T1T ) should be weak. However, if the relaxation is induced by a macroscopic field (i.e. motion
−1 2 19 of vortices), then T1 scales with γ . For F, which has a γ approximately four times higher than 13C, we do not observe an 19F signal with a wait time of approximately 2 hours
13 −1 in the temperature range where C(T1T ) enhancement occurs. Furthermore, due to a
−1 very narrow temperature and field range the (T1T ) enhancement effect resides, common relaxation mechanisms including the Hebel-Slichter peak in s-wave SCs and spin fluctuation can also be ruled out as described in [36].
EF
#
±"BB normalized DOS Density of State
Energy (k ) EnergyB
Figure 5.9: Sketch of density of states of a d-wave SC with nodes in the modulation of order parameter. ∆ here is the magnitude of superconducting gap
5.3.3 Picture of a fluctuating FFLO state
Another possible relaxation mechanism is the motion of the real space nodes that produces fluctuation magnetic field at the nuclear sites. We consider the specific case of an in-plane
−1 magnetic field, where (T1T ) enhancement of 35% higher than normal state at around B = 10 ∼ 11 T is observed. This will agree with our description of nodal motion if the modulation is pinned at a lower temperature and then thermally activated as temperature increases until it reaches the thermal dynamic boundary that destroys the modulation. That is, the motion of the nodes, which induces the fluctuating field that facilitate relaxation, is in
70 −1 a slow limit (τc << ωLarmor). Ideally, solid evidence proving a motional induced relaxation is provided by looking at the field dependence, equivalently frequency dependence, which is what we did to study the motion of ethylene group in Section 4.4. However, in this case, the effect of increasing the applied magnetic field on the magnitude of fluctuation field is complicated. It suppresses the magnitude of modulation, and simultaneously, the period of modulation is also reduced [71]. A consequence of increasing applied magnetic field is that the thermal dynamic phase boundary is pushed toward lower temperatures, leading
−1 to vanishing (T1T ) enhancement because the slow motion of the nodal plane is not fast enough to be observed before the order parameter is destroyed completely [46].
The picture of moving nodal planes is further supported if one considers the nature of the phase transition from normal to FFLO state. The symmetry is broken by a continuous modulation of order parameters, or equivalently, quasiparticle density, of which the modu- lation period is a function of field and temperature. In 1975, Imry and Ma developed an elegant argument describing the instability of long-range ordering under continuous sym- metry breaking [99]. They said that for a dimension d < 4 under random fluctuation field induced by disorders, the low symmetry phase that continuously breaks the symmetry is unstable toward domain formation and does not form long-range ordering. That is, for a system with disorder, the FFLO order parameter does not exhibit long-range order (i.e. a fluctuating FFLO state). This argument supports our picture of a moving nodal plane in the sense that short range modulation easily causes pinning of the nodes, leading to the
−1 motional (T1T ) enhancement in a slow limit, which is also consistent with specific heat measurements wherein a broad transition was reported [46].
−1 Although we have concluded that the enhancement in (T1T ) is not from the motion of vortices directly in Section 3.3.1, we believe vortex dynamics may be related to the en- hanced NMR relaxation indirectly by facilitating the motion of the nodal plane, and have to be incorporated to understand the out-of-plane enhancement. We start with a qualitative description of the effect of a magnetic field on the vortex dynamic. For out-of-plane magnetic fields, higher field leads to a higher density of vortices. The vortices tend to be more mobile at higher density due to stiffer vortex lattice. For the in-plane magnetic field, Josephson
71 coupling between layers becomes weaker at higher applied magnetic fields, and therefore, the Josephson vortices between layers are decoupled [100, 101]. Moreover, the nodal planes in the modulation of order parameters in the FFLO state, of which the motion may lead to enhanced NMR relaxation, were, in fact, reported to couple to the motion of vortices through measuring interlayer resistivity [100, 102]. For the in-plane condition at T = 749
−1 mK, (T1T ) enhancement maximizes at B = 10 T, because that may be the field required to reduce interlayer coupling to free the motion of the nodal plane. At B = 9.25 T, the vortices are pinned when the applied magnetic field is in-plane. As shown by the interlayer resistivity measurement, the critical field to melt the vortices’ structure drops rapidly as the magnetic field is tilted from the conduction plane [101]. Consequently, a huge enhancement is observed at 1◦ for B = 9.25 T in Figure 5.6. At a fixed field and increasing temperatures, the out-of-plane enhancement is suppressed and the maximum occurs at a lower angle, as demonstrated in Figure 5.6 (right). In this case, the enhancement is not only limited by the frequency of the motion, but also the thermal dynamic boundary that shrinks sharply to a lower temperature, as exhibited in specific heat measurement under a tilted field [46].
1300 0° 1200 0.75° 5° ) ) 1100 -1 -1
1000 (sec (sec 2 2 T T
1/ 900 1/
800
700 0.4 0.6 0.8 1 2 3 4 5 TemperatureTemperature (K) (K)
−1 Figure 5.10: Temperature dependence of T2 at B = 10.5 T
When the magnetic field fluctuates slowly, another powerful probe in NMR, the spin–
−1 + spin relaxation rate T2 , can be utilized. In the normal state T → Tc , T2 is of the order of approximately 1 ms. When the characteristic time scale of the fluctuation is comparable
−1 to T2, an enhancement occurs in T2 because of the boosted phase decoherence in nuclear
72 Larmor precession that kills the transverse nuclear magnetization. In Figure 5.10, the tem-
−1 perature dependence of T2 is shown, for a series of angles at B = 10.5 T. An increase in −1 ◦ T2 is observed for in-plane and 0.75 misaligned conditions, and completely suppressed for ◦ −1 5 misalignment. The enhancement of T2 certainly indicates the existence of a fluctuation −1 field near the parameter space where T1 enhancement is observed, which is qualitatively −1 consistent with the picture of (T1T ) enhancement induced by a fluctuation field.
−1 −1 Here, a quantitative approach to investigate whether the enhancements in T1 and T2 −1 are of the same physics is demonstrated. A motional-triggered enhancement of T2 is de- scribed by
2 −0.5 1/T2 ∼ 2/τc ;(τc >> h∆ω i – fast limit)
2 2 −0.5 1/T2 ∼ 2 < ∆ω > τc ;(τc << h∆ω i – slow limit)
h∆ω2i ∼ (hγ)2
where τc and h are the correlation time and local modulating field, respectively [103]. When
2 −0.5 −1 2 0.5 τc → h∆ω i , T2 reaches its maximum ∼ 2h∆ω i . The question here is whether −1 the peak in T2 in Figure 5.10 is originated from a thermally-activated motional effect. −1 3 −1 −3 Considering the enhancement of T2 ∼ 10 sec , τc will be ∼ 10 sec, and the effect of −1 −1 h∆ω2i −9 −1 this fluctuating field on T will be negligible with T ∼ 2 ∼ 10 sec . What about 1 1 ω τc −1 −1 in the case of the enhancement of T2 in fast or slow limit of T2 ? In the slow limit, 2 −0.5 −1 −1 where τc >> h∆ω i , the temperature dependence of T1 and T2 are consistent with a −3 fluctuating field h ∼ 385 G as shown in Figure 5.11 where correlation time τc ∼ 10 sec. −1 The enhancement of T2 in Figure 5.10 is cut off by thermaldynamic phase boundary, similar −1 to the temperature dependence of T1 . A question arises here: Is a modulating field h ∼ 385 G reasonable? In [55], a modeling of β”-SC spectrum in the FFLO state showed 1D order parameter modulation consistent with a dense domain wall picture and modulating field of
−1 the order of ∼ 10 G. Is it possible to have the modulating field seen in NMR relaxation T1 but not spectrum? This will only be the case if the spins at the domain walls are considered as isolated electrons so that the modulating field decays rapidly away from the domain walls instead of oscillating sinusoidally as described in [104]. However, the length scale that the
73 modulation can confine is at the order of ξ ∼ 15 nm, certainly not isolated. Alternatively
2 −0.5 −1 in a fast limit where τc << h∆ω i , the enhancement of T2 will agree with a fluctuating −7 −1 field h ∼ 4 G and τc ∼ 10 sec, whereas the effect on T1 remains the same. This fast −1 −1 limit will lead to a discrepancy of T2 at lower temperature where T2 seems to freeze out rapidly below 1 K. In short, we find that qualitatively a modulating field does result in the
−1 −1 −1 enhancement of T1 and T2 . However, for T2 , neither a picture of slow or fast limit in −1 a motional induced T2 is comprehensive, which indicates that the origin of enhancement −1 of T2 may be more complicated then pure hyperfine-origin, and, for example, the effect of macroscopic fluctuating field from vortices may come into play.
0.03 2000 Baseline Baseline Assisted by modulation 1800 Slow limit )
) 0.025
-1 Fast limit -1 )
) 1600 -1 -1 sec sec 0.02 1400 -1 -1 (sec (sec (K 2 2
) (K 1200 T T 0.015 T T 1 1 1/ 1/ T T 1000 1/
1/( 0.01 800
0.005 600 0.4 0.6 0.8 1 2 3 4 5 0.4 0.6 0.8 1 2 3 4 5 TemperatureTemperature (K) (K) TemperatureTemperature (K) (K)
−1 Figure 5.11: Modeling of the effect of modulated order parameter on (T1T ) (left) and −1 −3 T2 (right). Modulating field h, τ0 and Ea for slow or fast limit are set at ∼ 385 or 4 G, 10 or 10−7 sec and 1.5 K, respectively
−1 In summary, we argue that the enhancement in (T1T ) of in-plane and out-of-plane ap- plied magnetic fields in the FFLO state of layered SC are of the same origin. Moreover, while
−1 this enhancement of (T1T ) in NMR indicates the existence of nodes of order parameter modulation in the FFLO state, it should be interpreted as the motion of nodes facilitated by decoupling between layers at higher or tilted applied magnetic fields, rather than static bound states.
74 CHAPTER 6
Conclusion
Our NMR study reveals that β”-SC may be an example of a charge-fluctuation mediated unconventional SC, unlike most of other unconventional SCs which the pairing mechanism is commonly attributed to magnetic fluctuation. Temperature dependence of 13C NMR
−1 (T1T ) is consistent with an unconventional SC with nodal superconducting gap. Above Tc, little to no evidence of spin fluctuation is observed, and Korringa ratio analysis shows weak antiferromagnetic correlation compared to most of other unconventional SCs with magnetic ground state. Moreover, charge disproportionation is found between inequivalent molecular sites and intensifies upon cooling, where electron correlation may be enhanced by the motion of ethylene group at T > 100 K. At high pressure ∼ 8 kbar, charge disproportionation is suppressed in accordance with the drop in Tc previously shown in transport measurement [1], indicating possible interplay between charge ordering and superconductivity. Considering the origin of superconductivity, our result certainly calls for investigations of a direct evidence of charge fluctuation in β”-SC.
Furthermore, at low temperature 13C NMR study exposes interesting microscopic nature of the FFLO state. We observe evidence that is consistent with a fluctuating FFLO state depicted by short-range modulation of which the motions are coupled to the vortices, and are accelerated at high magnetic field, or when the magnetic field is misaligned from the
−1 −1 conducting plane. In both case, the motional enhanced (T1T ) is in a slow limit – τc << −1 ωLarmor. We propose that the enhancement of (T1T ) of β”-SC in the FFLO state, both in-plane and out-of-plane, are of hyperfine origin and should be attributed to the motion of the nodes in the modulation of superconducting order parameter.
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