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*What is "Spin Liquid" and what did we learn about it with NMR (Nulcear Magnetic Resonance)?

* What follows is a description of the concept of spin liquid in layperson's language, and is not meant to be scientifically rigorous.

1) What is the "spin" and how do they interact with each other?

Bar magnets, such as those used for a magnetic compass, have a North pole ("N") and a South pole ("S"): the N pole of a bar magnet is attracted by the S pole of another magnet.  On the other hand, the N pole repels another N pole (and of course the S pole repels another S pole).  The mutual attraction or repulsion between bar magnets is caused by magnetic fields generated by them ("dipole magnetic field").

In analogy with the macroscopic example of bar magnets above, each electron is known to have the equivalent of "N" and "S" at a microscopic level, and behaves like a tiny bar magnet.  Such magnetic properties of electrons may be described by a mathematical entity called a "spin."  Intuitively, one might imagine each electron comes with a tiny arrow (see figure below).  The head of the arrow (spin) maybe viewed as the N pole of the electron. 


When multiple electrons are combined within one atom, they can also behave like a small magnet.  For example, Cu2+ ions have 9 electrons in the 3d orbitals, and hence one electron is left over as an "unpaired electron."  It is the spin of this unpaired electron that gives rise to the magnetic properties of a Cu2+ ion.

2) What is ferromagnet and antiferromagnet?

When two spins are close to each other in a material, they interact with each other.  The mechanism of the interaction involves quantum mechanics and is more complicated than the case of the dipole magnetic fields between two bar magnets explained above; long story short, when there is a "ferromagnetic exchange interaction" between two adjacent spins, they would try to be parallel, and form a "ferromagnet" (left panel below).  Actually, the usual bar magnet is nothing but a ferromagnet consisting of a gigantic array of parallel electron spins, resulting in the formation of the macroscopic N and S poles.  On the other hand, two spins try to be anti-parallel under the presence of an "antiferromagnetic super-exchange interaction" that commonly exists when two Cu2+ ions are bonded to each other across an O2- ion.  For example, in the lanthanum-copper-oxide La2CuO4, Cu2+ ions form a square-lattice and O2- are located in between two Cu2+ ions, so the Cu2+ electron spins interact via the antiferromagnetic super-exchange interaction.  In antiferromagnets, spins try to form a regular pattern of "up" and "down" configurations (right panel below).  [If we remove some electrons from La2CuO4, it becomes the famous copper-oxide high temperature superconductor, but we are not getting into the details here.  An important point here, though, is that exotic superconductivity often manifests itself as a side effect of exotic magnetism exhibited by quantum spins; so achieving deep understanding of exotic superconductivity cannot be accomplished without achieving good understanding of the magnetic properties of the material.]  

ferro antiferro

When the temperature of
La2CuO4 is warm enough, thermal fluctuations beat the tendency of spins to become anti-parallel.  Accordingly, spins at Cu2+ ions point random orientations and keep fluctuating; this state is called "paramagnet" (left panel below).  When one cools down La2CuO4 to below about 50 degrees Celsius (about 320 K), however, thermal fluctuations are no longer strong enough to disrupt the collective will of all the spins to form an anti-parallel configuration.  As a consequence, a phase transition takes place, and an antiferromagnet (a.k.a. Neel state) is formed.  In the Neel state, all spins form a rigid, regular array of "up" and "down" configurations (right panel below); notice that each "up" spin is surrounded by "down" spins, and vice versa.


The transformation of the spin state from a paramagnet to a Neel state is an example of "phase transitions" in nature.  In a phase transition, a matter changes its form; for example, when we cool liquid water to below 0 degrees C, the liquid state of water undergoes a phase transition into a solid state, commonly know as "ice."  In the liquid state of water, H2O molecules freely move around, whereas water molecules form a rigid array of lattice in ice.  "Paramagnetic state" of spins and antiferromagnetic "Neel state" correspond to (crudely speaking) water (liquid) and ice, respectively.   


3) Frustration effects and "Spin Liquid"

In the early 20th century, L. Landau (Nobel Prize in Physics 1962) suggested that, theoretically, electron spins cannot form an antiferromagnetic (Neel) state due to quantum mechanical effects, and spins keep fluctuating, without forming a rigid regular pattern.  His proposal was largely forgotten after L. Neel experimentally demonstrated the existence of the Neel state (Nobel Prize in Physics 1972).  

In 1973, P. W. Anderson (Nobel Prize in Physics 1977) came up with a new idea.  He proposed that paramagnetic spins may not undergo a phase transition into a Neel's antiferromagnetic state under the presence of "frustration" effects, and form a "spin liquid;" spins keep fluctuating without forming a regular pattern in a spin liquid.  To understand his key idea, imagine Cu2+ and O2- ions forming a triangle as shown below.  Because of the aforementioned antiferromagnetic super-exchange interactions, two spins at adjacent Cu2+ ions 1 and 2 would try to form an "up" and "down" configuration.  When up and down spins are combined, they average to zero --- physicists call such a pair "singlet."  But then what would the third spin 3 at another corner do?  If the spin 3 is pointing down (left panel below), spin 1 and 3 are also mutually happy with each other, because they are anti-parallel.  On the other hand, spin 2 and spin 3 are parallel and hence hate each other.  To be more precise, the interaction energy between spin 2 and 3 becomes higher and frustrate them, destabilizing the relationship among the three spins. 


If spin 3 is "up" instead (right panel above), spin 2 and spin 3 are anti-parallel (that is good), but spin 1 and spin 3 end up becoming parallel, which is energetically unfavorable.  So, three electron spins form a "love triangle," so to speak, and we cannot make all three of them happy at once, resulting in continued fluctuations of spins even at absolute zero.  Stated differently, all spins may continuously alter the partner to form a singlet.  In such a frustrated situation,  spins remain in a fluid-like state ("spin liquid") rather than a solid-like state with fixed orientations ("Neel state").

4) Hunt for a spin liquid material

Since the 1970's, many solid state physicists and chemists have been looking for a frustrated spin system that exhibits the signature of a spin liquid state.  Among all the possible avenues to realize a spin liquid state, the kagome lattice has been considered one of the most promising candidates.  "Kagome" is a Japanese word meaning the "basket pattern", and consists of a network of corner-shared triangles (see an example of Japanese basket below).


Electron spins forming an infinite kagome network of corner-shared triangles might host a spin liquid state for the following reason.  Note that the kagome lattice in two panels shown below are filled with singlet pairs, with different patterns.  Since all the spins are involved in a singlet, they all cancel out, and hence the total magnitude of spin is zero in these patterns.  Moreover, these two configurations are energetically equivalent --- this also means that spins have difficulties in making up their mind to settle down with a particular pattern, and hence they may keep fluctuating by forming different patterns with the total spin zero, i.e. a spin liquid.


In 1972, a copper mineral material "herbertsmithite" ZnCu3(OH)
6Cl2 was discovered in a mine in Chili.  Herbertsmithite contains a layer of Cu2+ and O2- ions forming a kagome lattice as shown below [1].  Notice that Cu2+ ions (represented by blue balls) are bridged by O2- ions (represented by red balls), and the antiferromagnetic super-exchange interaction between the adjacent Cu2+ ions across the intervening O2- ions try to make spins form anti-parallel configurations.  Each Cu3O3 triangle is linked with three adjacent Cu3O3 triangles, forming the "kagome" pattern.  Synthesizing a purified form of ZnCu3(OH)6Cl2 in the laboratory, however, proved very difficult.  In 2005, a research group led by a chemist, D. Nocera (then at MIT, now at Harvard) succeeded in synthesizing ZnCu3(OH)6Cl2 artificially in the lab [1], and its physical properties were investigated by a research group led by Young Lee (then at MIT, now at Stanford) and many others using neutron scattering techniques etc.

kagome structure

Earlier experimental studies established that Cu2+ spins in herbertsmithite do not undergo a phase transition into a Neel state.  Furthermore, the spin excitations in
herbertsmithite do not take the form of typical "spin waves" that are commonly observed for Neel's antiferromagnets (Prof. B. Brockhouse of McMaster invented an experimental technique to observe spin waves, and won a Nobel Physics Prize in 1994).  These results strongly suggested that a spin liquid state may be indeed formed in the kagome lattice of herbertsmithite [2].  Unfortunately, probing the intrinsic low energy properties of Cu2+ spins in herbertsmithite is not easy based on experimental techniques that measure the average properties of a sample, because additional defect spins occupy the non-magnetic Zn2+ sites outside the kagome plane with 15% probability; their response dominates the low energy properties of the sample near absolute zero.  Thus one needs to find a way to probe the properties of Cu2+ spins on the kagome plane near absolute zero, separately from the response of the defect spins at Zn2+ sitesThat is where NMR (Nuclear Magnetic Resonance) techniques become handy.  In our cryogenic NMR measurements, we investigated the behavior of 17O nuclei in our herbertsmithite sample.  We can probe the intrinsic properties of Cu2+ spins on the kagome lattice by observing the behavior of 17O nuclei that are located far from the defect spins, because these 17O nuclei are primarily influenced by Cu2+ spins within the kagome plane in their vicinity and the presence of the defect spins hardly matters.  (We can also probe the properties of defect spins separately by observing the behavior of 17O nuclei sitting next to the former.  This site-selective nature of NMR is our powerful weapon.) 17O nuclei 17O nucl

17O single crystal NMR investigation of herbertsmithite               

Imai Lab has the expertise in conducting NMR (Nuclear Magnetic Resonance) measurements at low temperatures near absolute zero, and is specialized in investigating exotic quantum phenomena exhibited by electrons in solids. 
17O single crystal NMR project began in 2012 as a collaborative research project between Takashi Imai group at McMaster and Young Lee group at MIT (Lee has moved to Stanford since then).  NMR measurements became the subject of Ph.D. dissertation by a graduate student Mingxuan Fu in Imai group, and Tian-heng Han, then a Ph.D. student of Young Lee at MIT, grew a single crystal using 17O isotope enriched water as a starting ingredient (Han is now a postdoctoral fellow at Univ. of Chicago and Argonne National Lab).

NMR (Nuclear Magnetic Resonance) technique has a distinct advantage over many other experimental techniques, in that we can probe the physical properties of different locations within a sample separately as explained above.  MRI in hospitals utilizes this useful property of NMR technique to pinpoint the location of cancer, broken bones, etc. (MRI machines have an NMR spectrometer in it).   

Brockhouse Institute for Materials Research of McMaster (named after McMaster's own Prof. B. Brockouse, Nobel Physics Prize in 1994) has long traditions in providing supports for advanced materials research; thanks to the cryogenic support, we were able to cool  herbertsmithite down to about 1 kelvin (-272 degrees C; absolute 0 is -273.15 degrees C).  The X-ray diffraction user support service also helped us align the single crystal sample along a proper orientation.  These user supports are essential for ensuring that McMaster researchers can conduct world class research in quantum condensed matter physics. 

The graph below shows the temperature dependence of the Cu2+ spin susceptibility inferred from the 17O NMR frequency shift.  Spin susceptibility is a measure of how large the total spin of the kagome plane is.  It turned out that the intrinsic spin susceptibility asymptotes to zero near absolute zero ---- note that the data points drop down to zero near the lower left corner of the plot.  This implies that all spins are averaged out to zero by forming a network of fluctuating singlets --- the key, smoking gun evidence for the formation of a spin liquid state in the kagome lattice that everybody has been looking for for decades.  Moreover, from the measurements conducted in different magnetic fields, we were able to demonstrate that there is an energy gap in the excitation spectrum of spins.  Combined together, we were able to establish the existence of a gapped spin liquid ground state in the kagome lattice of herbertsmithite. 

Our discoveries based on NMR experiments were published in journal Science in the November 6 issue in 2015 [3].

kagome chi

[1] M. P. Shore, D. G. Nocera et al., J. Am. Chem. Soc. 127 (2005) 13462.
[2] Tian-heng Han, Young S. Lee et al., Nature 492 (2012) 406.
[3] Mingxuan Fu, Takashi Imai, Tian-heng Han, and Young S. Lee, Science 350 (2015) 655.