Physics:Quantum oscillations - Revision history

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	==Underdoped cuprates==		==Underdoped cuprates==
	Studies of underdoped cuprate compounds such as YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6+''x''</sub> through probes such as ARPES have indicated that these phases show characteristics of non-Fermi liquids,<ref name=alexandrov>{{cite journal\|last=Alexandrov\|first=A. S.\|title=Theory of quantum magneto-oscillations in underdoped cuprate superconductors\|journal=Journal of Physics: Condensed Matter\|year=2008\|volume=20\|issue=19\|article-number=192202\|doi=10.1088/0953-8984/20/19/192202\|bibcode = 2008JPCM...20s2202A \|arxiv = 0711.0093 \|s2cid=117020227}}</ref> and in particular, the absence of well-defined Landau quasiparticles.<ref name=rmp-arpes>{{cite journal\|last1=Damascelli\|first1=Andrea\|last2=Hussain\|first2=Zahid\|author3=Zhi-Xun Shen\|title=Angle-resolved photoemission studies of the cuprate superconductors\|journal=Reviews of Modern Physics\|year=2003\|volume=75\|issue=2\|page=473\|doi=10.1103/RevModPhys.75.473\|arxiv = cond-mat/0208504 \|bibcode = 2003RvMP...75..473D \|s2cid=118433150}}</ref> However, quantum oscillations have been observed in these materials at low temperatures, if their superconductivity is suppressed by a sufficiently high magnetic field,<ref name=leyraud /> which is evidence for the presence of well-defined quasiparticles with fermionic statistics. These experimental results thus disagree with those from ARPES and other probes.<ref name=sebastian2011 />		Studies of underdoped cuprate compounds such as YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6+''x''</sub> through probes such as ARPES have indicated that these phases show characteristics of non-Fermi liquids,<ref name=alexandrov>{{cite journal\|last=Alexandrov\|first=A. S.\|title=Theory of quantum magneto-oscillations in underdoped cuprate superconductors\|journal=Journal of Physics: Condensed Matter\|year=2008\|volume=20\|issue=19\|article-number=192202\|doi=10.1088/0953-8984/20/19/192202\|bibcode = 2008JPCM...20s2202A \|arxiv = 0711.0093 \|s2cid=117020227}}</ref> and in particular, the absence of well-defined Landau quasiparticles.<ref name=rmp-arpes>{{cite journal\|last1=Damascelli\|first1=Andrea\|last2=Hussain\|first2=Zahid\|author3=Zhi-Xun Shen\|title=Angle-resolved photoemission studies of the cuprate superconductors\|journal=Reviews of Modern Physics\|year=2003\|volume=75\|issue=2\|page=473\|doi=10.1103/RevModPhys.75.473\|arxiv = cond-mat/0208504 \|bibcode = 2003RvMP...75..473D \|s2cid=118433150}}</ref> However, quantum oscillations have been observed in these materials at low temperatures, if their superconductivity is suppressed by a sufficiently high magnetic field,<ref name=leyraud /> which is evidence for the presence of well-defined quasiparticles with fermionic statistics. These experimental results thus disagree with those from ARPES and other probes.<ref name=sebastian2011 />

	~~==See also==~~
	* de Haas–van Alphen effect
	* Shubnikov–de Haas effect
	* Landau levels

	== See also ==		== See also ==

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	In condensed matter physics, '''quantum oscillations''' describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field.<ref name="prsa">{{cite journal\|last=Coldea\|first=Amalia\|year=2010\|title=Quantum oscillations probe the normal electronic states of novel superconductors\|url=http://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|journal=Philosophical Transactions of the Royal Society A\|volume=368\|issue=1924\|pages=3503–3517\|bibcode=2010RSPTA.368.3503C\|doi=10.1098/rsta.2010.0089\|pmid=20603364\|access-date=20 March 2012\|doi-access=free\|url-access=subscription\|archive-date=21 April 2023\|archive-url=https://web.archive.org/web/20230421121056/https://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|url-status=dead}}</ref> These techniques are based on the principle of Landau quantization of Fermions moving in a magnetic field.<ref name=leyraud>{{cite journal\|last=Doiron-Leyraud\|first=Nicolas\|title=Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor\|journal=Nature \|year=2007\|volume=447\|doi=10.1038/nature05872\|bibcode = 2007Natur.447..565D \|arxiv = 0801.1281 \|pmid=17538614 \|issue=7144 \|pages=565–8\|s2cid=4397560\|display-authors=0}}</ref> For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the ''Landau levels'', whose separation is proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic density of states at the Fermi level; this produces oscillations in the many material properties which depend on this, including resistance (the Shubnikov–de Haas effect), [[Physics:Quantum Hall effect\|Hall resistance]],<ref name=leyraud /> and magnetic susceptibility (the de Haas–van Alphen effect). Observation of quantum oscillations in a material is considered a signature of Fermi liquid behaviour.<ref name=nrc>{{cite book\|title=Condensed-matter and materials physics: the science of the world around us\|year=2010\|publisher=National Research Council\|isbn=978-0-309-13409-5\|url=https://books.google.com/books?id=_50wVzbzDzkC&q=Sr2RhO4+quantum+oscillation+fermi+liquid&pg=PT60}}</ref>		In condensed matter physics, '''quantum oscillations''' describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field.<ref name="prsa">{{cite journal\|last=Coldea\|first=Amalia\|year=2010\|title=Quantum oscillations probe the normal electronic states of novel superconductors\|url=http://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|journal=Philosophical Transactions of the Royal Society A\|volume=368\|issue=1924\|pages=3503–3517\|bibcode=2010RSPTA.368.3503C\|doi=10.1098/rsta.2010.0089\|pmid=20603364\|access-date=20 March 2012\|doi-access=free\|url-access=subscription\|archive-date=21 April 2023\|archive-url=https://web.archive.org/web/20230421121056/https://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|url-status=dead}}</ref> These techniques are based on the principle of Landau quantization of Fermions moving in a magnetic field.<ref name=leyraud>{{cite journal\|last=Doiron-Leyraud\|first=Nicolas\|title=Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor\|journal=Nature \|year=2007\|volume=447\|doi=10.1038/nature05872\|bibcode = 2007Natur.447..565D \|arxiv = 0801.1281 \|pmid=17538614 \|issue=7144 \|pages=565–8\|s2cid=4397560\|display-authors=0}}</ref> For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the ''Landau levels'', whose separation is proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic density of states at the Fermi level; this produces oscillations in the many material properties which depend on this, including resistance (the Shubnikov–de Haas effect), [[Physics:Quantum Hall effect\|Hall resistance]],<ref name=leyraud /> and magnetic susceptibility (the de Haas–van Alphen effect). Observation of quantum oscillations in a material is considered a signature of Fermi liquid behaviour.<ref name=nrc>{{cite book\|title=Condensed-matter and materials physics: the science of the world around us\|year=2010\|publisher=National Research Council\|isbn=978-0-309-13409-5\|url=https://books.google.com/books?id=_50wVzbzDzkC&q=Sr2RhO4+quantum+oscillation+fermi+liquid&pg=PT60}}</ref>

	Quantum oscillations have been used to study high temperature superconducting materials such as ~~[[Physics:High-temperature superconductivity#Cuprates\|~~cuprates]] and pnictides.<ref name=prsa /> Studies using these experiments have shown that the ground state of underdoped cuprates behave similar to a Fermi liquid, and display characteristics such as Landau quasiparticles.<ref name=dmbroun>{{cite journal\|last=Broun\|first=D. M.\|title=What lies beneath the dome?\|journal=Nature Physics\|year=2008\|volume=4\|pages=170–172\|bibcode = 2008NatPh...4..170B \|doi = 10.1038/nphys909\|issue=3}}</ref>		Quantum oscillations have been used to study high temperature superconducting materials such as cuprates and pnictides.<ref name=prsa /> Studies using these experiments have shown that the ground state of underdoped cuprates behave similar to a Fermi liquid, and display characteristics such as Landau quasiparticles.<ref name=dmbroun>{{cite journal\|last=Broun\|first=D. M.\|title=What lies beneath the dome?\|journal=Nature Physics\|year=2008\|volume=4\|pages=170–172\|bibcode = 2008NatPh...4..170B \|doi = 10.1038/nphys909\|issue=3}}</ref>

	In 2021 this technique has been used to observe a predicted state called "electron–phonon fluid",<ref>{{Cite journal\|last1=Yang\|first1=Hung-Yu\|last2=Yao\|first2=Xiaohan\|last3=Plisson\|first3=Vincent\|last4=Mozaffari\|first4=Shirin\|last5=Scheifers\|first5=Jan P.\|last6=Savvidou\|first6=Aikaterini Flessa\|last7=Choi\|first7=Eun Sang\|last8=McCandless\|first8=Gregory T.\|last9=Padlewski\|first9=Mathieu F.\|last10=Putzke\|first10=Carsten\|last11=Moll\|first11=Philip J. W.\|date=2021-09-06\|title=Evidence of a coupled electron-phonon liquid in NbGe2\|journal=Nature Communications\|language=en\|volume=12\|issue=1\|page=5292\|doi=10.1038/s41467-021-25547-x\|pmid=34489411\|issn=2041-1723\|pmc=8421384\|arxiv=2103.01515\|bibcode=2021NatCo..12.5292Y}}</ref><ref>{{Cite web\|last=College\|first=Boston\|date=2021-09-06\|title=Novel Metal Discovered Where Electrons Flow in the Same Way Water Flows in a Pipe\|url=https://scitechdaily.com/novel-metal-discovered-where-electrons-flow-in-the-same-way-water-flows-in-a-pipe/\|access-date=2021-09-20\|website=SciTechDaily\|language=en-US}}</ref> a similar particle-quasiparticle state already known is the exciton–polariton fluid.		In 2021 this technique has been used to observe a predicted state called "electron–phonon fluid",<ref>{{Cite journal\|last1=Yang\|first1=Hung-Yu\|last2=Yao\|first2=Xiaohan\|last3=Plisson\|first3=Vincent\|last4=Mozaffari\|first4=Shirin\|last5=Scheifers\|first5=Jan P.\|last6=Savvidou\|first6=Aikaterini Flessa\|last7=Choi\|first7=Eun Sang\|last8=McCandless\|first8=Gregory T.\|last9=Padlewski\|first9=Mathieu F.\|last10=Putzke\|first10=Carsten\|last11=Moll\|first11=Philip J. W.\|date=2021-09-06\|title=Evidence of a coupled electron-phonon liquid in NbGe2\|journal=Nature Communications\|language=en\|volume=12\|issue=1\|page=5292\|doi=10.1038/s41467-021-25547-x\|pmid=34489411\|issn=2041-1723\|pmc=8421384\|arxiv=2103.01515\|bibcode=2021NatCo..12.5292Y}}</ref><ref>{{Cite web\|last=College\|first=Boston\|date=2021-09-06\|title=Novel Metal Discovered Where Electrons Flow in the Same Way Water Flows in a Pipe\|url=https://scitechdaily.com/novel-metal-discovered-where-electrons-flow-in-the-same-way-water-flows-in-a-pipe/\|access-date=2021-09-20\|website=SciTechDaily\|language=en-US}}</ref> a similar particle-quasiparticle state already known is the exciton–polariton fluid.

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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	In ~~[[Physics:Condensed matter physics\|~~condensed matter physics]], '''quantum oscillations''' describes a series of related ~~[[Physics:Experimental physics\|~~experimental]] techniques used to map the ~~[[Physics:Fermi surface\|~~Fermi surface]] of a ~~[[Chemistry:Metal\|~~metal]] in the presence of a strong ~~[[Magnetic field\|~~magnetic field]].<ref name="prsa">{{cite journal\|last=Coldea\|first=Amalia\|year=2010\|title=Quantum oscillations probe the normal electronic states of novel superconductors\|url=http://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|journal=Philosophical Transactions of the Royal Society A\|volume=368\|issue=1924\|pages=3503–3517\|bibcode=2010RSPTA.368.3503C\|doi=10.1098/rsta.2010.0089\|pmid=20603364\|access-date=20 March 2012\|doi-access=free\|url-access=subscription\|archive-date=21 April 2023\|archive-url=https://web.archive.org/web/20230421121056/https://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|url-status=dead}}</ref> These techniques are based on the principle of ~~[[Physics:Landau quantization\|~~Landau quantization]] of Fermions moving in a magnetic field.<ref name=leyraud>{{cite journal\|last=Doiron-Leyraud\|first=Nicolas\|title=Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor\|journal=Nature \|year=2007\|volume=447\|doi=10.1038/nature05872\|bibcode = 2007Natur.447..565D \|arxiv = 0801.1281 \|pmid=17538614 \|issue=7144 \|pages=565–8\|s2cid=4397560\|display-authors=0}}</ref> For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the ''Landau levels'', whose separation is proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic ~~[[Physics:Density of states\|~~density of states]] at the ~~[[Physics:Fermi level\|~~Fermi level]]; this produces oscillations in the many material properties which depend on this, including resistance (the ~~[[Physics:~~Shubnikov–de Haas effect~~\|Shubnikov–de Haas effect]]~~), [[Physics:Quantum Hall effect\|Hall resistance]],<ref name=leyraud /> and ~~[[Physics:Magnetic susceptibility\|~~magnetic susceptibility]] (the ~~[[Physics:De Haas–van Alphen effect\|~~de Haas–van Alphen effect]]). Observation of quantum oscillations in a material is considered a signature of ~~[[Physics:Fermi liquid theory\|~~Fermi liquid]] behaviour.<ref name=nrc>{{cite book\|title=Condensed-matter and materials physics: the science of the world around us\|year=2010\|publisher=National Research Council\|isbn=978-0-309-13409-5\|url=https://books.google.com/books?id=_50wVzbzDzkC&q=Sr2RhO4+quantum+oscillation+fermi+liquid&pg=PT60}}</ref>		In condensed matter physics, '''quantum oscillations''' describes a series of related experimental techniques used to map the Fermi surface of a metal in the presence of a strong magnetic field.<ref name="prsa">{{cite journal\|last=Coldea\|first=Amalia\|year=2010\|title=Quantum oscillations probe the normal electronic states of novel superconductors\|url=http://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|journal=Philosophical Transactions of the Royal Society A\|volume=368\|issue=1924\|pages=3503–3517\|bibcode=2010RSPTA.368.3503C\|doi=10.1098/rsta.2010.0089\|pmid=20603364\|access-date=20 March 2012\|doi-access=free\|url-access=subscription\|archive-date=21 April 2023\|archive-url=https://web.archive.org/web/20230421121056/https://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html\|url-status=dead}}</ref> These techniques are based on the principle of Landau quantization of Fermions moving in a magnetic field.<ref name=leyraud>{{cite journal\|last=Doiron-Leyraud\|first=Nicolas\|title=Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor\|journal=Nature \|year=2007\|volume=447\|doi=10.1038/nature05872\|bibcode = 2007Natur.447..565D \|arxiv = 0801.1281 \|pmid=17538614 \|issue=7144 \|pages=565–8\|s2cid=4397560\|display-authors=0}}</ref> For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the ''Landau levels'', whose separation is proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic density of states at the Fermi level; this produces oscillations in the many material properties which depend on this, including resistance (the Shubnikov–de Haas effect), [[Physics:Quantum Hall effect\|Hall resistance]],<ref name=leyraud /> and magnetic susceptibility (the de Haas–van Alphen effect). Observation of quantum oscillations in a material is considered a signature of Fermi liquid behaviour.<ref name=nrc>{{cite book\|title=Condensed-matter and materials physics: the science of the world around us\|year=2010\|publisher=National Research Council\|isbn=978-0-309-13409-5\|url=https://books.google.com/books?id=_50wVzbzDzkC&q=Sr2RhO4+quantum+oscillation+fermi+liquid&pg=PT60}}</ref>

	Quantum oscillations have been used to study high temperature superconducting materials such as [[Physics:High-temperature superconductivity#Cuprates\|cuprates]] and pnictides.<ref name=prsa /> Studies using these experiments have shown that the ground state of ~~[[Physics:Doping (semiconductor)\|~~underdoped]] cuprates behave similar to a Fermi liquid, and display characteristics such as Landau ~~[[Physics:Quasiparticle\|quasiparticle]]s~~.<ref name=dmbroun>{{cite journal\|last=Broun\|first=D. M.\|title=What lies beneath the dome?\|journal=Nature Physics\|year=2008\|volume=4\|pages=170–172\|bibcode = 2008NatPh...4..170B \|doi = 10.1038/nphys909\|issue=3}}</ref>		Quantum oscillations have been used to study high temperature superconducting materials such as [[Physics:High-temperature superconductivity#Cuprates\|cuprates]] and pnictides.<ref name=prsa /> Studies using these experiments have shown that the ground state of underdoped cuprates behave similar to a Fermi liquid, and display characteristics such as Landau quasiparticles.<ref name=dmbroun>{{cite journal\|last=Broun\|first=D. M.\|title=What lies beneath the dome?\|journal=Nature Physics\|year=2008\|volume=4\|pages=170–172\|bibcode = 2008NatPh...4..170B \|doi = 10.1038/nphys909\|issue=3}}</ref>

	In 2021 this technique has been used to observe a predicted state called "electron–phonon fluid",<ref>{{Cite journal\|last1=Yang\|first1=Hung-Yu\|last2=Yao\|first2=Xiaohan\|last3=Plisson\|first3=Vincent\|last4=Mozaffari\|first4=Shirin\|last5=Scheifers\|first5=Jan P.\|last6=Savvidou\|first6=Aikaterini Flessa\|last7=Choi\|first7=Eun Sang\|last8=McCandless\|first8=Gregory T.\|last9=Padlewski\|first9=Mathieu F.\|last10=Putzke\|first10=Carsten\|last11=Moll\|first11=Philip J. W.\|date=2021-09-06\|title=Evidence of a coupled electron-phonon liquid in NbGe2\|journal=Nature Communications\|language=en\|volume=12\|issue=1\|page=5292\|doi=10.1038/s41467-021-25547-x\|pmid=34489411\|issn=2041-1723\|pmc=8421384\|arxiv=2103.01515\|bibcode=2021NatCo..12.5292Y}}</ref><ref>{{Cite web\|last=College\|first=Boston\|date=2021-09-06\|title=Novel Metal Discovered Where Electrons Flow in the Same Way Water Flows in a Pipe\|url=https://scitechdaily.com/novel-metal-discovered-where-electrons-flow-in-the-same-way-water-flows-in-a-pipe/\|access-date=2021-09-20\|website=SciTechDaily\|language=en-US}}</ref> a similar particle-quasiparticle state already known is the ~~[[Physics:Polariton superfluid\|~~exciton–polariton fluid]].		In 2021 this technique has been used to observe a predicted state called "electron–phonon fluid",<ref>{{Cite journal\|last1=Yang\|first1=Hung-Yu\|last2=Yao\|first2=Xiaohan\|last3=Plisson\|first3=Vincent\|last4=Mozaffari\|first4=Shirin\|last5=Scheifers\|first5=Jan P.\|last6=Savvidou\|first6=Aikaterini Flessa\|last7=Choi\|first7=Eun Sang\|last8=McCandless\|first8=Gregory T.\|last9=Padlewski\|first9=Mathieu F.\|last10=Putzke\|first10=Carsten\|last11=Moll\|first11=Philip J. W.\|date=2021-09-06\|title=Evidence of a coupled electron-phonon liquid in NbGe2\|journal=Nature Communications\|language=en\|volume=12\|issue=1\|page=5292\|doi=10.1038/s41467-021-25547-x\|pmid=34489411\|issn=2041-1723\|pmc=8421384\|arxiv=2103.01515\|bibcode=2021NatCo..12.5292Y}}</ref><ref>{{Cite web\|last=College\|first=Boston\|date=2021-09-06\|title=Novel Metal Discovered Where Electrons Flow in the Same Way Water Flows in a Pipe\|url=https://scitechdaily.com/novel-metal-discovered-where-electrons-flow-in-the-same-way-water-flows-in-a-pipe/\|access-date=2021-09-20\|website=SciTechDaily\|language=en-US}}</ref> a similar particle-quasiparticle state already known is the exciton–polariton fluid.
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	==Experiment==		==Experiment==
	When a magnetic field is applied to a system of free charged ~~[[Physics:Fermion\|~~fermions]], their energy states are quantized into the so-called Landau levels, given by<ref name=sebastian2011>{{cite journal\|last=Sebastian\|first=Suchitra E.\|author2=Neil Harrison\|author3=Gilbert G. Lonzarich\|title=Quantum oscillations in the high-Tc cuprates\|journal=Philosophical Transactions of the Royal Society A\|year=2011\|volume=369\|pages=1687–1711\|doi=10.1098/rsta.2010.0243\|pmid=21422021\|url=http://rsta.royalsocietypublishing.org/content/369/1941/1687.full.pdf+html\|access-date=23 March 2012\|bibcode=2011RSPTA.369.1687S\|issue=1941\|doi-access=free\|url-access=subscription\|archive-date=17 April 2023\|archive-url=https://web.archive.org/web/20230417054936/https://rsta.royalsocietypublishing.org/content/369/1941/1687.full.pdf+html\|url-status=dead}}</ref>		When a magnetic field is applied to a system of free charged fermions, their energy states are quantized into the so-called Landau levels, given by<ref name=sebastian2011>{{cite journal\|last=Sebastian\|first=Suchitra E.\|author2=Neil Harrison\|author3=Gilbert G. Lonzarich\|title=Quantum oscillations in the high-Tc cuprates\|journal=Philosophical Transactions of the Royal Society A\|year=2011\|volume=369\|pages=1687–1711\|doi=10.1098/rsta.2010.0243\|pmid=21422021\|url=http://rsta.royalsocietypublishing.org/content/369/1941/1687.full.pdf+html\|access-date=23 March 2012\|bibcode=2011RSPTA.369.1687S\|issue=1941\|doi-access=free\|url-access=subscription\|archive-date=17 April 2023\|archive-url=https://web.archive.org/web/20230417054936/https://rsta.royalsocietypublishing.org/content/369/1941/1687.full.pdf+html\|url-status=dead}}</ref>
	[[File:Quantum oscillations at 100 T.jpg\|180px\|thumb\|right\|[[YBCO]] superconductor under high magnetic field. As field strength is increased, superconductivity is suppressed and Landau oscillations can be observed]]		[[File:Quantum oscillations at 100 T.jpg\|180px\|thumb\|right\|YBCO superconductor under high magnetic field. As field strength is increased, superconductivity is suppressed and Landau oscillations can be observed]]
	<math>\varepsilon_l=\frac{eB}{m^*}\left(\ell +\frac{1}{2}\right)</math>		<math>\varepsilon_l=\frac{eB}{m^*}\left(\ell +\frac{1}{2}\right)</math>

	for integer-valued <math>\ell</math>, where <math>B</math> is the external magnetic field and <math>e,m^*</math> are the fermion charge and ~~[[Physics:Effective mass (solid-state physics)\|~~effective mass]] respectively.		for integer-valued <math>\ell</math>, where <math>B</math> is the external magnetic field and <math>e,m^*</math> are the fermion charge and effective mass respectively.

	When the external magnetic field <math>B</math> is increased in an isolated system, the Landau levels expand, and eventually "fall off" the Fermi surface. This leads to oscillations in the observed energy of the highest occupied level, and hence in many physical properties (including Hall conductivity, resistivity, and susceptibility). The periodicity of these oscillations can be measured, and in turn can be used to determine the cross-sectional area of the Fermi surface.<ref name=ibachnluth>{{cite book\|last=Ibach\|first=Harald\|author2=Hans Lüth\|title=Solid-state physics: an introduction to principles of materials science\|year=1995\|publisher=Springer-Verlag\|location=Berlin\|isbn=978-3-540-58573-2\|url=https://books.google.com/books?id=PIEfweaKyK8C&q=%22quantum+oscillations%22+magnetic+experimental+techniques+in+physics&pg=PA260}}</ref> If the axis of the magnetic field is varied at constant magnitude, similar oscillations are observed. The oscillations occur whenever the Landau orbits touch the Fermi surface. In this way, the complete geometry of the Fermi sphere can be mapped.<ref name=ibachnluth />		When the external magnetic field <math>B</math> is increased in an isolated system, the Landau levels expand, and eventually "fall off" the Fermi surface. This leads to oscillations in the observed energy of the highest occupied level, and hence in many physical properties (including Hall conductivity, resistivity, and susceptibility). The periodicity of these oscillations can be measured, and in turn can be used to determine the cross-sectional area of the Fermi surface.<ref name=ibachnluth>{{cite book\|last=Ibach\|first=Harald\|author2=Hans Lüth\|title=Solid-state physics: an introduction to principles of materials science\|year=1995\|publisher=Springer-Verlag\|location=Berlin\|isbn=978-3-540-58573-2\|url=https://books.google.com/books?id=PIEfweaKyK8C&q=%22quantum+oscillations%22+magnetic+experimental+techniques+in+physics&pg=PA260}}</ref> If the axis of the magnetic field is varied at constant magnitude, similar oscillations are observed. The oscillations occur whenever the Landau orbits touch the Fermi surface. In this way, the complete geometry of the Fermi sphere can be mapped.<ref name=ibachnluth />

	==Underdoped cuprates==		==Underdoped cuprates==
	Studies of underdoped cuprate compounds such as YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6+''x''</sub> through probes such as ARPES have indicated that these phases show characteristics of non-Fermi liquids,<ref name=alexandrov>{{cite journal\|last=Alexandrov\|first=A. S.\|title=Theory of quantum magneto-oscillations in underdoped cuprate superconductors\|journal=Journal of Physics: Condensed Matter\|year=2008\|volume=20\|issue=19\|article-number=192202\|doi=10.1088/0953-8984/20/19/192202\|bibcode = 2008JPCM...20s2202A \|arxiv = 0711.0093 \|s2cid=117020227}}</ref> and in particular, the absence of well-defined Landau ~~[[Physics:Quasiparticle\|quasiparticle]]s~~.<ref name=rmp-arpes>{{cite journal\|last1=Damascelli\|first1=Andrea\|last2=Hussain\|first2=Zahid\|author3=Zhi-Xun Shen\|title=Angle-resolved photoemission studies of the cuprate superconductors\|journal=Reviews of Modern Physics\|year=2003\|volume=75\|issue=2\|page=473\|doi=10.1103/RevModPhys.75.473\|arxiv = cond-mat/0208504 \|bibcode = 2003RvMP...75..473D \|s2cid=118433150}}</ref> However, quantum oscillations have been observed in these materials at low temperatures, if their superconductivity is suppressed by a sufficiently high magnetic field,<ref name=leyraud /> which is evidence for the presence of well-defined quasiparticles with ~~[[Physics:Fermi–Dirac statistics\|~~fermionic statistics]]. These experimental results thus disagree with those from ARPES and other probes.<ref name=sebastian2011 />		Studies of underdoped cuprate compounds such as YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6+''x''</sub> through probes such as ARPES have indicated that these phases show characteristics of non-Fermi liquids,<ref name=alexandrov>{{cite journal\|last=Alexandrov\|first=A. S.\|title=Theory of quantum magneto-oscillations in underdoped cuprate superconductors\|journal=Journal of Physics: Condensed Matter\|year=2008\|volume=20\|issue=19\|article-number=192202\|doi=10.1088/0953-8984/20/19/192202\|bibcode = 2008JPCM...20s2202A \|arxiv = 0711.0093 \|s2cid=117020227}}</ref> and in particular, the absence of well-defined Landau quasiparticles.<ref name=rmp-arpes>{{cite journal\|last1=Damascelli\|first1=Andrea\|last2=Hussain\|first2=Zahid\|author3=Zhi-Xun Shen\|title=Angle-resolved photoemission studies of the cuprate superconductors\|journal=Reviews of Modern Physics\|year=2003\|volume=75\|issue=2\|page=473\|doi=10.1103/RevModPhys.75.473\|arxiv = cond-mat/0208504 \|bibcode = 2003RvMP...75..473D \|s2cid=118433150}}</ref> However, quantum oscillations have been observed in these materials at low temperatures, if their superconductivity is suppressed by a sufficiently high magnetic field,<ref name=leyraud /> which is evidence for the presence of well-defined quasiparticles with fermionic statistics. These experimental results thus disagree with those from ARPES and other probes.<ref name=sebastian2011 />

	==See also==		==See also==
	* ~~[[Physics:De Haas–van Alphen effect\|~~de Haas–van Alphen effect]]		* de Haas–van Alphen effect
	* ~~[[Physics:Shubnikov–de Haas effect\|~~Shubnikov–de Haas effect]]		* Shubnikov–de Haas effect
	* ~~[[Physics:Landau levels\|~~Landau levels]]		* Landau levels

	== See also ==		== See also ==

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	* [[Physics:Shubnikov–de Haas effect\|Shubnikov–de Haas effect]]		* [[Physics:Shubnikov–de Haas effect\|Shubnikov–de Haas effect]]
	* [[Physics:Landau levels\|Landau levels]]		* [[Physics:Landau levels\|Landau levels]]

			== See also ==
			{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	==References==		==References==
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	[[Category:Magnetism]]		[[Category:Magnetism]]

	{{Sourceattribution\|Quantum oscillations}}		{{Author\|Harold Foppele}}

			{{Sourceattribution\|Physics:Quantum oscillations\|1}}

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 {{Quantum book backlink|Quantum dynamics and evolution}}
 In [[Physics:Condensed matter physics|condensed matter physics]], '''quantum oscillations''' describes a series of related [[Physics:Experimental physics|experimental]] techniques used to map the [[Physics:Fermi surface|Fermi surface]] of a [[Chemistry:Metal|metal]] in the presence of a strong [[Magnetic field|magnetic field]].<ref name="prsa">{{cite journal|last=Coldea|first=Amalia|year=2010|title=Quantum oscillations probe the normal electronic states of novel superconductors|url=http://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html|journal=Philosophical Transactions of the Royal Society A|volume=368|issue=1924|pages=3503–3517|bibcode=2010RSPTA.368.3503C|doi=10.1098/rsta.2010.0089|pmid=20603364|access-date=20 March 2012|doi-access=free|url-access=subscription|archive-date=21 April 2023|archive-url=https://web.archive.org/web/20230421121056/https://rsta.royalsocietypublishing.org/content/368/1924/3503.full.pdf+html|url-status=dead}}</ref> These techniques are based on the principle of [[Physics:Landau quantization|Landau quantization]] of Fermions moving in a magnetic field.<ref name=leyraud>{{cite journal|last=Doiron-Leyraud|first=Nicolas|title=Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor|journal=Nature |year=2007|volume=447|doi=10.1038/nature05872|bibcode = 2007Natur.447..565D |arxiv = 0801.1281 |pmid=17538614 |issue=7144 |pages=565–8|s2cid=4397560|display-authors=0}}</ref> For a gas of free fermions in a strong magnetic field, the energy levels are quantized into bands, called the ''Landau levels'', whose separation is proportional to the strength of the magnetic field. In a quantum oscillation experiment, the external magnetic field is varied, which causes the Landau levels to pass over the Fermi surface, which in turn results in oscillations of the electronic [[Physics:Density of states|density of states]] at the [[Physics:Fermi level|Fermi level]]; this produces oscillations in the many material properties which depend on this, including resistance (the [[Physics:Shubnikov–de Haas effect|Shubnikov–de Haas effect]]), [[Physics:Quantum Hall effect|Hall resistance]],<ref name=leyraud /> and [[Physics:Magnetic susceptibility|magnetic susceptibility]] (the [[Physics:De Haas–van Alphen effect|de Haas–van Alphen effect]]). Observation of quantum oscillations in a material is considered a signature of [[Physics:Fermi liquid theory|Fermi liquid]] behaviour.<ref name=nrc>{{cite book|title=Condensed-matter and materials physics: the science of the world around us|year=2010|publisher=National Research Council|isbn=978-0-309-13409-5|url=https://books.google.com/books?id=_50wVzbzDzkC&q=Sr2RhO4+quantum+oscillation+fermi+liquid&pg=PT60}}</ref>
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 In 2021 this technique has been used to observe a predicted state called "electron–phonon fluid",<ref>{{Cite journal|last1=Yang|first1=Hung-Yu|last2=Yao|first2=Xiaohan|last3=Plisson|first3=Vincent|last4=Mozaffari|first4=Shirin|last5=Scheifers|first5=Jan P.|last6=Savvidou|first6=Aikaterini Flessa|last7=Choi|first7=Eun Sang|last8=McCandless|first8=Gregory T.|last9=Padlewski|first9=Mathieu F.|last10=Putzke|first10=Carsten|last11=Moll|first11=Philip J. W.|date=2021-09-06|title=Evidence of a coupled electron-phonon liquid in NbGe2|journal=Nature Communications|language=en|volume=12|issue=1|page=5292|doi=10.1038/s41467-021-25547-x|pmid=34489411|issn=2041-1723|pmc=8421384|arxiv=2103.01515|bibcode=2021NatCo..12.5292Y}}</ref><ref>{{Cite web|last=College|first=Boston|date=2021-09-06|title=Novel Metal Discovered Where Electrons Flow in the Same Way Water Flows in a Pipe|url=https://scitechdaily.com/novel-metal-discovered-where-electrons-flow-in-the-same-way-water-flows-in-a-pipe/|access-date=2021-09-20|website=SciTechDaily|language=en-US}}</ref> a similar particle-quasiparticle state already known is the [[Physics:Polariton superfluid|exciton–polariton fluid]].
 ==Experiment==