665:
2843:. Phonons are the main source of heat conductivity in materials, where optical phonons contribute far less than acoustic phonons. This is because of the relatively low group velocity of optical phonons. When the thickness of the material decreases, the conductivity of via acoustic also decreases, since surface scattering increases. This microelectronics are getting smaller and smaller, reductions is getting more problematic. Although optical phonons themselves do not have a high thermal conductivity, SPhPs do seem to have this. So they may be an alternative means of cooling these electronic devices.
42:
678:
2080:
787:
2888:(CW) of polariton, or with an ultrafast pulse, producing a polariton with a very high temporal footprint. In both cases the polaritons are detected by the tip of the AFM, this signal is then used to calculate the energy of the polariton. One can also perform these experiments near the edge of the sample. This will result in the polaritons being
2343:
1532:
2413:
Towards the right of the crossing point, the upper branch behaves like a photon. The physical interpretation of this effect is that the frequency becomes too high for the ions to partake in the vibration, causing them to be essentially static. This results in a dispersion relation resembling one of a
2380:
The solution of this dispersion relation has two branches, an upper branch and a lower branch (see also the figure). If the slope of the curve is low, the particle is said to behave "phononlike", and if the slope is high the particle behaves "photonlike", owing these names to the slopes of the
847:
A phonon polariton is a type of quasiparticle that can form in some crystals due to the coupling of photons and lattice vibrations. They have properties of both light and sound waves, and can travel at very slow speeds in the material. They are useful for manipulating electromagnetic fields at
1389:
2584:
1076:. The difference lies in the magnitudes of their speeds, the speed of photons is many times larger than the speed for the acoustic phonons. The dispersion relations will therefore never cross each other, resulting in a lack of coupling. The dispersion relations touch at
2718:
2182:
1751:
in matter. Since, macroscopically, the crystal is uncharged and there is no current, the equations can be simplified. A phonon polariton must abide all of these six equations. To find solutions to this set of equations, we write the following trial
2074:
1990:
1906:
2883:
The induction of polaritons is very similar to that in Raman experiments, with a few differences. With the extremely high special resolution of SNOM, one can induce polaritons very locally in the sample. This can be done continuously, producing a
1135:
and have a negative slope, which is also much smaller in magnitude to that of photons. This will result in the crossing of the optical phonon branch and the photon dispersion, leading to their coupling and the forming of a phonon polariton.
2405:
The dispersion relation describes the behaviour of the coupling. The coupling of the phonon and the photon is the most promininent in the region where the original transverse disperion relations would have crossed. In the limit of large
2831:
Surface phonon polariton(SPhPs) are a specific kind of phonon polariton. They are formed by the coupling of optical surface phonon, instead of normal phonons, and light, resulting in an electromagnetic surface wave. They are similar to
1395:
1257:
1199:
2465:
2930:. In this field phonon polaritons are used for high speed signal processing and terahertz spectroscopy. The real-space imaging of phonon polaritons was made possible by projecting them onto a CCD camera.
3058:
Ambrosio, Antonio; Jauregui, Luis A.; Dai, Siyuan; Chaudhary, Kundan; Tamagnone, Michele; Fogler, Michael M.; Basov, Dimitri N.; Capasso, Federico; Kim, Philip; Wilson, William L. (2017-09-26).
1144:
The behavior of the phonon polaritons can be described by the dispersion relation. This dispersion relation is most easily derived for diatomic ion crystals with optical isotropy, for example
934:
2176:. Solving the resulting equations for ω and k, the magnitude of the wave vector, yields the following dispersion relation, and furthermore an expression for the optical dielectric constant:
852:
of the phonon and photon and their interaction. Photons consist of electromagnetic waves, which are always transverse. Therefore, they can only couple with transverse phonons in crystals.
2629:
2821:
1569:
2338:{\displaystyle {\frac {k^{2}c^{2}}{\omega ^{2}}}=\epsilon _{\infty }+{\frac {\epsilon _{0}-\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}{\omega _{0}}^{2}=\epsilon (\omega )}
1066:
1655:
2896:
will be created, which will again be detected by the AFM tip. In the case of the polaritons created by the ultrafast laser, no standing wave will be created. The wave can still
2637:
2375:
2087:. Red curves are the uncoupled phonon and photon dispersion relations, black curves are the result of coupling (from top to bottom: upper polariton, LO phonon, lower polariton).
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1996:
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2170:
1527:{\displaystyle \mathbf {P} =({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {w} +({\frac {\epsilon _{\infty }-1}{4\pi }})\mathbf {E} }
848:
nanoscale and enhancing optical phenomena. Phonon polaritons only result from coupling of transverse optical phonons, this is due to the particular form of the
1152:. Since the atoms in the crystal are charged, any lattice vibration which changes the relative distance between the two atoms in the unit cell will change the
2900:
with itself the moment it is reflected of the edge. Whether one is observing on the bulk surface or close to an edge, the signal is in temporal form. One can
2836:, although studied to a far lesser extent. The applications are far ranging from materials with negative index of refraction to high-density IR data storage.
1384:{\displaystyle {\ddot {\mathbf {w} }}=-{\omega _{0}}^{2}\mathbf {w} +({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {E} }
883:
the dispersion relation of an acoustic phonon can be approximated as being linear, with a particular gradient giving a dispersion relation of the form
709:
2410:, the solid lines of both branches approach the dotted lines, meaning, the coupling does not have a large impact on the behaviour of the vibrations.
2459:
is defined by the zero of the equation for the dielectric constant. Writing the equation for the dielectric constant in a different way yields:
2851:
Most observations of phonon polaritons are made of surface phonon polaritons, since these can be easily probed by Raman spectroscopy or AFM.
2579:{\displaystyle \epsilon (\omega )={\frac {{\omega _{0}}^{2}\epsilon _{0}-\omega ^{2}\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}}
1162:
3532:
Wu, Y.; Ordonez-Miranda, J.; Gluchko, S.; Anufriev, R.; Meneses, D. De Sousa; Del Campo, L.; Volz, S.; Nomura, M. (2020-10-02).
754:. Phonon polaritons occur in the region where the wavelength and energy of phonons and photons are similar, as to adhere to the
766:
702:
3305:
2778:
2427:
2875:
and the known laser energy, one can calculate the polariton energy, with which one can construct the dispersion relation.
1251:
Using this parameter, the behavior of the lattice vibrations for long waves can be described by the following equations:
886:
3644:
Feurer, T.; Stoyanov, Nikolay S.; Ward, David W.; Vaughan, Joshua C.; Statz, Eric R.; Nelson, Keith A. (2007-08-01).
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834:
808:
695:
682:
17:
816:
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1156:
of the material. To describe these vibrations, it is useful to introduce the parameter w, which is given by:
2904:
this signal, converting the signal into frequency domain, which can used to obtain the dispersion relation.
1543:
2713:{\displaystyle {\frac {{\omega _{L}}^{2}}{{\omega _{0}}^{2}}}={\frac {\epsilon _{0}}{\epsilon _{\infty }}}}
622:
102:
1033:
2069:{\displaystyle \mathbf {E} =\mathbf {E_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}
1985:{\displaystyle \mathbf {P} =\mathbf {P_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}
1901:{\displaystyle \mathbf {w} =\mathbf {w_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}
1633:
2351:
971:
627:
252:
2949:
2833:
517:
192:
1110:
1079:
858:
3284:
Borstel, G.; Falge, H. J.; Otto, A. (1974), Bauer, G.; Borstel, G.; Falge, H. J.; Otto, A. (eds.),
2381:
regular dispersion curves for phonons and photons. The phonon polariton behaves phononlike for low
1599:
797:
723:
512:
507:
33:
2116:
2094:
1803:
1781:
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1724:
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939:
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3598:
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1664:
1153:
801:
770:
202:
2146:
607:
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1748:
592:
532:
502:
452:
172:
62:
3645:
3422:"Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation"
3060:"Mechanical Detection and Imaging of Hyperbolic Phonon Polaritons in Hexagonal Boron Nitride"
632:
247:
232:
3669:
2867:
is chosen, this laser can induce the formation of a polariton on the sample. Looking at the
3712:
3657:
3545:
3496:
3433:
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3339:
3160:
3032:
2982:
2889:
2414:
regular photon in a crystal. The lower branch in this region behaves, because of their low
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8:
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272:
122:
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3549:
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3164:
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3744:
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3626:
3574:
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3184:
3105:
3071:
2971:"Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals"
762:
642:
602:
577:
325:
316:
3020:
3736:
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3673:
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3579:
3561:
3514:
3457:
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3402:
3355:
3301:
3292:, Springer Tracts in Modern Physics, Berlin, Heidelberg: Springer, pp. 107–148,
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3610:
3597:
Yao, Ziheng; Xu, Suheng; Hu, Debo; Chen, Xinzhong; Dai, Qing; Liu, Mengkun (2020).
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2723:
This equation gives the ratio of the frequency of the longitudonal optical phonon (
755:
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242:
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167:
137:
97:
57:
2885:
1145:
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52:
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3232:
3180:
3093:
3002:
2994:
2893:
731:
348:
329:
311:
212:
132:
3724:
3533:
3375:"Photonic approach to making a material with a negative index of refraction"
3286:"Surface and bulk phonon-polaritons observed by attenuated total reflection"
3270:
3085:
1030:. The dispersion relation of photons is also linear, being also of the form
773:(AFM) have made it possible to observe the polaritons in a more direct way.
542:
3740:
3614:
3583:
3557:
3484:
3461:
3101:
2919:
2913:
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2389:
in the lower branch. Conversely, the polariton behaves photonlike for high
1238:
1149:
562:
552:
522:
482:
477:
457:
302:
282:
142:
3599:"Nanoimaging and Nanospectroscopy of Polaritons with Time Resolved s-SNOM"
3509:
3123:
3059:
1747:
For the full coupling between the phonon and the photon, we need the four
3328:"Near-field imaging of mid-infrared surface phonon polariton propagation"
3250:
2927:
647:
582:
557:
527:
472:
467:
399:
41:
3327:
3297:
2864:
1753:
1229:
is the displacement of the positive atom relative to the negative atom;
492:
334:
127:
3421:
3351:
3148:
3445:
3285:
3172:
2939:
2923:
747:
547:
497:
370:
217:
117:
3485:"Thermal Conductivity of Amorphous Vs Crystalline Ge and GeTe Films"
3326:
Huber, A.; Ocelic, N.; Kazantsev, D.; Hillenbrand, R. (2005-08-22).
2079:
786:
3076:
2418:
compared to the photons, as regular transverse lattice vibrations.
1107:
Optical phonons, by contrast, have a non-zero angular frequency at
107:
1194:{\displaystyle \mathbf {w} =\mathbf {q} {\sqrt {\frac {\mu }{V}}}}
735:
427:
412:
375:
366:
361:
3531:
3699:
Feurer, T.; Vaughan, Joshua C.; Nelson, Keith A. (2003-01-17).
3325:
2944:
761:
Phonon polariton spectra have traditionally been studied using
743:
739:
380:
356:
87:
3701:"Spatiotemporal Coherent Control of Lattice Vibrational Waves"
3207:"Dynamical Theory of Crystal Lattices by M. Born and K. Huang"
1104:, but since the waves have no energy, no coupling will occur.
2860:
2823:
can be found using inelastic neutron scattering experiments.
2400:
751:
385:
82:
3057:
3534:"Enhanced thermal conduction by surface phonon-polaritons"
3643:
2863:
is pointed at the material being studied. If the correct
92:
3149:"Lattice Vibrations and Optical Waves in Ionic Crystals"
2777:) in diatomic cubic ionic crystals, and is known as the
2907:
2750:), to the frequency of the transverse optical phonon (
2172:
should be perpendicular to the electric field and the
2787:
2756:
2729:
2640:
2595:
2468:
2438:
2354:
2185:
2149:
2119:
2097:
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1806:
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1698:
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1602:
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1546:
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1213:
1165:
1113:
1082:
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974:
942:
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1022:
992:
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3283:
2918:Phonon polaritons also find use in the field of
2421:
929:{\displaystyle \omega _{\rm {ac}}=v_{\rm {ac}}k}
2143:the angular frequency. Notice that wave vector
27:Quasiparticle form phonon and photon coupling
3018:
703:
3019:Henry, C. H.; Hopfield, J. J. (1965-12-20).
2083:Dispersion relation of phonon polaritons in
3596:
2846:
2839:One other application is in the cooling of
2826:
2113:denotes the wave vector of the plane wave,
815:. Unsourced material may be challenged and
765:. The recent advances in (scattering-type)
3482:
3420:Ocelic, N.; Hillenbrand, R. (2004-08-01).
2432:The longitudonal optical phonon frequency
2401:Limit behaviour of the dispersion relation
710:
696:
40:
3573:
3508:
3222:
3075:
1657:is the high-frequency dielectric constant
835:Learn how and when to remove this message
3483:Nath, Prem; Chopra, K. L. (1974-01-01).
2968:
2624:{\displaystyle \epsilon (\omega _{L})=0}
2078:
3670:10.1146/annurev.matsci.37.052506.084327
2816:{\displaystyle \omega _{L}/\omega _{0}}
14:
3761:
3372:
3248:
3204:
1571:denotes the double time derivative of
1564:{\displaystyle {\ddot {\mathbf {w} }}}
1139:
1008:the absolute value of the wave vector
767:scanning near-field optical microscopy
3146:
2854:
3255:(8th ed.). Hoboken, NJ: Wiley.
3244:
3242:
3200:
3198:
3014:
3012:
1688:is the infrared dispersion frequency
1061:{\displaystyle \omega _{\rm {p}}=ck}
813:adding citations to reliable sources
780:
3650:Annual Review of Materials Research
3489:Japanese Journal of Applied Physics
3252:Introduction to solid state physics
2908:Polaritonics and real-space imaging
2871:emitted radiation and by using the
1650:{\displaystyle \epsilon _{\infty }}
24:
3124:"Phonon-polaritons | Nelson Group"
2703:
2534:
2385:in the upper branch, and for high
2370:{\displaystyle \epsilon (\omega )}
2259:
2230:
1642:
1494:
1431:
1332:
1043:
993:{\displaystyle \omega _{\rm {ac}}}
984:
981:
952:
949:
917:
914:
899:
896:
734:that can form in a diatomic ionic
25:
3780:
3239:
3195:
3009:
2377:the optical dielectric constant.
3021:"Raman Scattering by Polaritons"
2892:. In the case of CW polaritons,
2859:As with any Raman experiment, a
2121:
2099:
2040:
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2014:
2010:
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3525:
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3413:
3366:
3205:Wilson, A. J. C. (1955-07-01).
2878:
1743:is the dielectric polarization.
1247:is the volume of the unit cell.
3373:Shvets, Gennady (2003-01-16).
3319:
3277:
3140:
3116:
3051:
2969:Hopfield, J. J. (1958-12-01).
2962:
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2472:
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2358:
2332:
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2156:
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2028:
1969:
1944:
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1860:
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1483:
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1407:
1349:
1308:
1128:{\displaystyle \mathbf {k} =0}
1097:{\displaystyle \mathbf {k} =0}
876:{\displaystyle \mathbf {k} =0}
746:. They are particular type of
13:
1:
2955:
2779:Lyddane-Sachs-Teller relation
2428:Lyddane–Sachs–Teller relation
2422:Lyddane–Sachs–Teller relation
1616:{\displaystyle \epsilon _{0}}
2128:{\displaystyle \mathbf {x} }
2106:{\displaystyle \mathbf {k} }
1815:{\displaystyle \mathbf {P} }
1793:{\displaystyle \mathbf {E} }
1771:{\displaystyle \mathbf {w} }
1736:{\displaystyle \mathbf {P} }
1707:{\displaystyle \mathbf {E} }
1586:{\displaystyle \mathbf {w} }
1222:{\displaystyle \mathbf {q} }
1023:{\displaystyle \mathbf {k} }
961:{\displaystyle v_{\rm {ac}}}
7:
3147:HUANG, KUN (May 12, 1951).
2933:
2770:{\displaystyle \omega _{0}}
2743:{\displaystyle \omega _{L}}
2452:{\displaystyle \omega _{L}}
1681:{\displaystyle \omega _{0}}
10:
3785:
3603:Advanced Optical Materials
3399:10.1103/physrevb.67.035109
3045:10.1103/PhysRevLett.15.964
2911:
2834:surface plasmon polaritons
2425:
2165:{\displaystyle {\vec {k}}}
756:avoided crossing principle
740:transverse optical phonons
253:Spin gapless semiconductor
3224:10.1107/s0365110x5500279x
2950:Surface plasmon polariton
2855:Raman spectroscopy
2393:in the upper branch, low
776:
193:Electronic band structure
3646:"Terahertz Polaritonics"
3249:Kittel, Charles (2005).
2995:10.1103/physrev.112.1555
2847:Experimental observation
2827:Surface phonon polariton
1074:speed of light in vacuum
724:condensed matter physics
103:Bose–Einstein condensate
34:Condensed matter physics
3725:10.1126/science.1078726
3332:Applied Physics Letters
3086:10.1021/acsnano.7b02323
3025:Physical Review Letters
1154:dielectric polarization
968:the speed of the wave,
771:atomic force microscopy
3615:10.1002/adom.201901042
3558:10.1126/sciadv.abb4461
3211:Acta Crystallographica
2873:conservation of energy
2817:
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3510:10.7567/jjaps.2s1.781
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2589:Solving the equation
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2397:in the lower branch.
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248:Topological insulator
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1111:
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809:improve this section
750:, which behave like
266:Electronic phenomena
113:Fermionic condensate
3717:2003Sci...299..374F
3662:2007AnRMS..37..317F
3550:2020SciA....6.4461W
3501:1974JJAPS..13..781N
3438:2004NatMa...3..606O
3391:2003PhRvB..67c5109S
3344:2005ApPhL..87h1103H
3290:Solid-State Physics
3165:1951Natur.167..779H
3037:1965PhRvL..15..964H
2987:1958PhRv..112.1555H
1749:Maxwell's equations
1625:dielectric constant
1140:Dispersion relation
850:dispersion relation
738:due to coupling of
273:Quantum Hall effect
3298:10.1007/bfb0041387
2922:, a field between
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763:Raman spectroscopy
670:Physics portal
3711:(5605): 374–377.
3379:Physical Review B
3352:10.1063/1.2032595
3307:978-3-540-37868-6
3159:(4254): 779–780.
2902:Fourier transform
2708:
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2220:
2159:
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1980:
1896:
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1241:of the two atoms;
1189:
1188:
1002:angular frequency
845:
844:
837:
720:
719:
418:Granular material
186:Electronic phases
18:Phonon polaritons
16:(Redirected from
3776:
3753:
3752:
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3690:
3689:
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3635:
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3594:
3588:
3587:
3577:
3538:Science Advances
3529:
3523:
3522:
3512:
3480:
3474:
3473:
3446:10.1038/nmat1194
3426:Nature Materials
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3113:
3079:
3070:(9): 8741–8746.
3055:
3049:
3048:
3016:
3007:
3006:
2981:(5): 1555–1567.
2966:
2841:microelectronics
2822:
2820:
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2250:
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2209:
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2112:
2110:
2109:
2104:
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2075:
2073:
2072:
2067:
2065:
2062:
2057:
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2043:
2035:
2019:
2018:
2017:
2004:
1991:
1989:
1988:
1983:
1981:
1978:
1973:
1972:
1959:
1951:
1935:
1934:
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1920:
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1570:
1568:
1567:
1562:
1560:
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1549:
1533:
1531:
1530:
1525:
1523:
1515:
1513:
1505:
1498:
1497:
1487:
1479:
1474:
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1464:
1463:
1459:
1446:
1444:
1436:
1435:
1434:
1422:
1421:
1411:
1403:
1390:
1388:
1387:
1382:
1380:
1375:
1374:
1365:
1364:
1360:
1347:
1345:
1337:
1336:
1335:
1323:
1322:
1312:
1304:
1299:
1298:
1293:
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1291:
1274:
1273:
1268:
1263:
1228:
1226:
1225:
1220:
1218:
1200:
1198:
1197:
1192:
1190:
1181:
1180:
1178:
1170:
1134:
1132:
1131:
1126:
1118:
1103:
1101:
1100:
1095:
1087:
1067:
1065:
1064:
1059:
1048:
1047:
1046:
1029:
1027:
1026:
1021:
1019:
999:
997:
996:
991:
989:
988:
987:
967:
965:
964:
959:
957:
956:
955:
935:
933:
932:
927:
922:
921:
920:
904:
903:
902:
882:
880:
879:
874:
866:
840:
833:
829:
826:
820:
789:
781:
728:phonon polariton
712:
705:
698:
685:
680:
679:
672:
668:
667:
278:Spin Hall effect
168:Phase transition
138:Luttinger liquid
75:States of matter
58:Phase transition
44:
30:
29:
21:
3784:
3783:
3779:
3778:
3777:
3775:
3774:
3773:
3759:
3758:
3757:
3756:
3697:
3693:
3642:
3638:
3595:
3591:
3530:
3526:
3481:
3477:
3418:
3414:
3371:
3367:
3324:
3320:
3312:
3310:
3308:
3282:
3278:
3263:
3247:
3240:
3203:
3196:
3145:
3141:
3132:
3130:
3122:
3121:
3117:
3056:
3052:
3031:(25): 964–966.
3017:
3010:
2975:Physical Review
2967:
2963:
2958:
2936:
2916:
2910:
2886:continuous wave
2881:
2857:
2849:
2829:
2807:
2803:
2798:
2792:
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2786:
2783:
2782:
2761:
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2734:
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2725:
2724:
2702:
2698:
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2688:
2686:
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2663:
2662:
2656:
2649:
2645:
2644:
2643:
2641:
2639:
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2606:
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2591:
2590:
2567:
2563:
2554:
2547:
2543:
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2510:
2506:
2500:
2493:
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2464:
2463:
2443:
2439:
2437:
2434:
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2424:
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2314:
2307:
2303:
2302:
2301:
2292:
2288:
2279:
2272:
2268:
2267:
2266:
2265:
2258:
2254:
2245:
2241:
2240:
2238:
2229:
2225:
2214:
2210:
2203:
2199:
2193:
2189:
2188:
2186:
2184:
2181:
2180:
2151:
2150:
2148:
2145:
2144:
2120:
2118:
2115:
2114:
2098:
2096:
2093:
2092:
2061:
2039:
2031:
2024:
2020:
2013:
2009:
2008:
2000:
1998:
1995:
1994:
1977:
1955:
1947:
1940:
1936:
1929:
1925:
1924:
1916:
1914:
1911:
1910:
1893:
1871:
1863:
1856:
1852:
1845:
1841:
1840:
1832:
1830:
1827:
1826:
1807:
1805:
1802:
1801:
1785:
1783:
1780:
1779:
1763:
1761:
1758:
1757:
1728:
1726:
1723:
1722:
1699:
1697:
1694:
1693:
1672:
1668:
1666:
1663:
1662:
1641:
1637:
1635:
1632:
1631:
1607:
1603:
1601:
1598:
1597:
1578:
1576:
1573:
1572:
1550:
1548:
1547:
1545:
1542:
1541:
1519:
1506:
1493:
1489:
1488:
1486:
1475:
1469:
1465:
1455:
1451:
1447:
1437:
1430:
1426:
1417:
1413:
1412:
1410:
1399:
1397:
1394:
1393:
1376:
1370:
1366:
1356:
1352:
1348:
1338:
1331:
1327:
1318:
1314:
1313:
1311:
1300:
1294:
1287:
1283:
1282:
1281:
1264:
1262:
1261:
1259:
1256:
1255:
1214:
1212:
1209:
1208:
1179:
1174:
1166:
1164:
1161:
1160:
1146:sodium chloride
1142:
1114:
1112:
1109:
1108:
1083:
1081:
1078:
1077:
1042:
1041:
1037:
1035:
1032:
1031:
1015:
1013:
1010:
1009:
980:
979:
975:
973:
970:
969:
948:
947:
943:
941:
938:
937:
913:
912:
908:
895:
894:
890:
888:
885:
884:
862:
860:
857:
856:
841:
830:
824:
821:
806:
790:
779:
769:((s-)SNOM) and
716:
675:
662:
661:
654:
653:
652:
442:
434:
433:
432:
408:Amorphous solid
402:
392:
391:
390:
369:
351:
341:
340:
339:
328:
326:Antiferromagnet
319:
317:Superparamagnet
310:
297:
296:Magnetic phases
289:
288:
287:
267:
259:
258:
257:
187:
179:
178:
177:
163:Order parameter
157:
156:Phase phenomena
149:
148:
147:
77:
67:
28:
23:
22:
15:
12:
11:
5:
3782:
3772:
3771:
3769:Quasiparticles
3755:
3754:
3691:
3656:(1): 317–350.
3636:
3609:(5): 1901042.
3589:
3524:
3475:
3432:(9): 606–609.
3412:
3365:
3318:
3306:
3276:
3261:
3238:
3194:
3139:
3128:nelson.mit.edu
3115:
3050:
3008:
2960:
2959:
2957:
2954:
2953:
2952:
2947:
2942:
2935:
2932:
2912:Main article:
2909:
2906:
2894:standing waves
2880:
2877:
2869:Stokes shifted
2856:
2853:
2848:
2845:
2828:
2825:
2810:
2806:
2801:
2795:
2791:
2764:
2760:
2737:
2733:
2721:
2720:
2705:
2701:
2695:
2691:
2685:
2678:
2671:
2667:
2659:
2652:
2648:
2620:
2617:
2614:
2609:
2605:
2601:
2598:
2587:
2586:
2570:
2566:
2562:
2557:
2550:
2546:
2536:
2532:
2526:
2522:
2518:
2513:
2509:
2503:
2496:
2492:
2483:
2480:
2477:
2474:
2471:
2446:
2442:
2426:Main article:
2423:
2420:
2416:phase velocity
2402:
2399:
2366:
2363:
2360:
2357:
2346:
2345:
2334:
2331:
2328:
2325:
2322:
2317:
2310:
2306:
2295:
2291:
2287:
2282:
2275:
2271:
2261:
2257:
2253:
2248:
2244:
2237:
2232:
2228:
2224:
2217:
2213:
2206:
2202:
2196:
2192:
2174:magnetic field
2158:
2155:
2139:the time, and
2135:the position,
2123:
2101:
2077:
2076:
2060:
2055:
2052:
2049:
2046:
2042:
2038:
2034:
2030:
2027:
2023:
2016:
2012:
2007:
2003:
1992:
1976:
1971:
1968:
1965:
1962:
1958:
1954:
1950:
1946:
1943:
1939:
1932:
1928:
1923:
1919:
1908:
1892:
1887:
1884:
1881:
1878:
1874:
1870:
1866:
1862:
1859:
1855:
1848:
1844:
1839:
1835:
1810:
1788:
1766:
1756:solutions for
1745:
1744:
1731:
1719:
1718:
1716:electric field
1702:
1690:
1689:
1675:
1671:
1659:
1658:
1644:
1640:
1628:
1627:
1623:is the static
1610:
1606:
1594:
1593:
1581:
1557:
1553:
1535:
1534:
1522:
1518:
1512:
1509:
1504:
1501:
1496:
1492:
1485:
1482:
1478:
1472:
1468:
1462:
1458:
1454:
1450:
1443:
1440:
1433:
1429:
1425:
1420:
1416:
1409:
1406:
1402:
1391:
1379:
1373:
1369:
1363:
1359:
1355:
1351:
1344:
1341:
1334:
1330:
1326:
1321:
1317:
1310:
1307:
1303:
1297:
1290:
1286:
1280:
1277:
1271:
1267:
1249:
1248:
1242:
1231:
1230:
1217:
1202:
1201:
1187:
1184:
1177:
1173:
1169:
1141:
1138:
1124:
1121:
1117:
1093:
1090:
1086:
1057:
1054:
1051:
1045:
1040:
1018:
986:
983:
978:
954:
951:
946:
925:
919:
916:
911:
907:
901:
898:
893:
872:
869:
865:
843:
842:
793:
791:
784:
778:
775:
718:
717:
715:
714:
707:
700:
692:
689:
688:
687:
686:
673:
656:
655:
651:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
515:
510:
505:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
444:
443:
440:
439:
436:
435:
431:
430:
425:
423:Liquid crystal
420:
415:
410:
404:
403:
398:
397:
394:
393:
389:
388:
383:
378:
373:
364:
359:
353:
352:
349:Quasiparticles
347:
346:
343:
342:
338:
337:
332:
323:
314:
308:Superdiamagnet
305:
299:
298:
295:
294:
291:
290:
286:
285:
280:
275:
269:
268:
265:
264:
261:
260:
256:
255:
250:
245:
240:
235:
233:Thermoelectric
230:
228:Superconductor
225:
220:
215:
210:
208:Mott insulator
205:
200:
195:
189:
188:
185:
184:
181:
180:
176:
175:
170:
165:
159:
158:
155:
154:
151:
150:
146:
145:
140:
135:
130:
125:
120:
115:
110:
105:
100:
95:
90:
85:
79:
78:
73:
72:
69:
68:
66:
65:
60:
55:
49:
46:
45:
37:
36:
26:
9:
6:
4:
3:
2:
3781:
3770:
3767:
3766:
3764:
3750:
3746:
3742:
3738:
3734:
3730:
3726:
3722:
3718:
3714:
3710:
3706:
3702:
3695:
3687:
3683:
3679:
3675:
3671:
3667:
3663:
3659:
3655:
3651:
3647:
3640:
3632:
3628:
3624:
3620:
3616:
3612:
3608:
3604:
3600:
3593:
3585:
3581:
3576:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3539:
3535:
3528:
3520:
3516:
3511:
3506:
3502:
3498:
3494:
3490:
3486:
3479:
3471:
3467:
3463:
3459:
3455:
3451:
3447:
3443:
3439:
3435:
3431:
3427:
3423:
3416:
3408:
3404:
3400:
3396:
3392:
3388:
3385:(3): 035109.
3384:
3380:
3376:
3369:
3361:
3357:
3353:
3349:
3345:
3341:
3338:(8): 081103.
3337:
3333:
3329:
3322:
3309:
3303:
3299:
3295:
3291:
3287:
3280:
3272:
3268:
3264:
3262:0-471-41526-X
3258:
3254:
3253:
3245:
3243:
3234:
3230:
3225:
3220:
3216:
3212:
3208:
3201:
3199:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3150:
3143:
3129:
3125:
3119:
3111:
3107:
3103:
3099:
3095:
3091:
3087:
3083:
3078:
3073:
3069:
3065:
3061:
3054:
3046:
3042:
3038:
3034:
3030:
3026:
3022:
3015:
3013:
3004:
3000:
2996:
2992:
2988:
2984:
2980:
2976:
2972:
2965:
2961:
2951:
2948:
2946:
2943:
2941:
2938:
2937:
2931:
2929:
2925:
2921:
2915:
2905:
2903:
2899:
2895:
2891:
2887:
2876:
2874:
2870:
2866:
2862:
2852:
2844:
2842:
2837:
2835:
2824:
2808:
2804:
2799:
2793:
2789:
2780:
2762:
2758:
2735:
2731:
2699:
2693:
2689:
2683:
2676:
2669:
2665:
2657:
2650:
2646:
2634:
2633:
2632:
2618:
2615:
2607:
2603:
2596:
2568:
2564:
2560:
2555:
2548:
2544:
2530:
2524:
2520:
2516:
2511:
2507:
2501:
2494:
2490:
2481:
2475:
2469:
2462:
2461:
2460:
2444:
2440:
2429:
2419:
2417:
2411:
2409:
2398:
2396:
2392:
2388:
2384:
2378:
2361:
2355:
2329:
2323:
2320:
2315:
2308:
2304:
2293:
2289:
2285:
2280:
2273:
2269:
2255:
2251:
2246:
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2025:
2021:
2005:
1993:
1974:
1966:
1963:
1960:
1952:
1941:
1937:
1921:
1909:
1890:
1882:
1879:
1876:
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1857:
1853:
1837:
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1824:
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1423:
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1367:
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794:This section
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783:
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759:
757:
753:
749:
745:
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737:
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732:quasiparticle
730:is a type of
729:
725:
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708:
706:
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699:
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693:
691:
690:
684:
674:
671:
666:
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646:
644:
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639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
614:
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609:
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604:
601:
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589:
586:
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581:
579:
576:
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559:
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536:
534:
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524:
521:
519:
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511:
509:
506:
504:
501:
499:
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494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
448:Van der Waals
446:
445:
438:
437:
429:
426:
424:
421:
419:
416:
414:
411:
409:
406:
405:
401:
396:
395:
387:
384:
382:
379:
377:
374:
372:
368:
365:
363:
360:
358:
355:
354:
350:
345:
344:
336:
333:
331:
327:
324:
322:
318:
315:
313:
309:
306:
304:
301:
300:
293:
292:
284:
281:
279:
276:
274:
271:
270:
263:
262:
254:
251:
249:
246:
244:
243:Ferroelectric
241:
239:
238:Piezoelectric
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
213:Semiconductor
211:
209:
206:
204:
201:
199:
196:
194:
191:
190:
183:
182:
174:
171:
169:
166:
164:
161:
160:
153:
152:
144:
141:
139:
136:
134:
133:Superfluidity
131:
129:
126:
124:
121:
119:
116:
114:
111:
109:
106:
104:
101:
99:
96:
94:
91:
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86:
84:
81:
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3127:
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3067:
3063:
3053:
3028:
3024:
2978:
2974:
2964:
2920:polaritonics
2917:
2914:Polaritonics
2882:
2879:SNOM and AFM
2858:
2850:
2838:
2830:
2781:. The ratio
2722:
2588:
2431:
2412:
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2390:
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1239:reduced mass
1234:
1203:
1150:zinc sulfide
1143:
1106:
1069:
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854:
846:
831:
822:
807:Please help
795:
760:
727:
721:
578:von Klitzing
283:Kondo effect
143:Time crystal
123:Fermi liquid
3495:(S1): 781.
2928:electronics
400:Soft matter
321:Ferromagnet
3313:2023-07-27
3217:(7): 444.
3133:2024-01-30
3077:1704.01834
2956:References
2865:wavelength
1754:plane wave
1072:being the
825:April 2022
543:Louis NĂ©el
533:Schrieffer
441:Scientists
335:Spin glass
330:Metamagnet
312:Paramagnet
128:Supersolid
3733:0036-8075
3678:1531-7331
3631:203134796
3623:2195-1071
3566:2375-2548
3519:0021-4922
3454:1476-1122
3407:0163-1829
3360:0003-6951
3233:0365-110X
3181:0028-0836
3094:1936-0851
3003:0031-899X
2940:Polariton
2924:photonics
2898:interfere
2890:reflected
2805:ω
2790:ω
2759:ω
2732:ω
2704:∞
2700:ϵ
2690:ϵ
2666:ω
2647:ω
2604:ω
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2305:ω
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2157:→
2048:ω
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2037:⋅
1964:ω
1961:−
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1670:ω
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1500:−
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1329:ϵ
1325:−
1316:ϵ
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1279:−
1270:¨
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1039:ω
977:ω
892:ω
796:does not
748:polariton
643:Wetterich
623:Abrikosov
538:Josephson
508:Van Vleck
498:Luttinger
371:Polariton
303:Diamagnet
223:Conductor
218:Semimetal
203:Insulator
118:Fermi gas
3763:Category
3749:19627306
3741:12532012
3686:33353438
3584:32998899
3470:21116893
3462:15286756
3271:55228781
3189:30926099
3102:28858472
3064:ACS Nano
2934:See also
2631:yields:
683:Category
628:Ginzburg
603:Laughlin
563:Kadanoff
518:Shockley
503:Anderson
458:von Laue
108:Bose gas
3713:Bibcode
3705:Science
3658:Bibcode
3575:7527230
3546:Bibcode
3497:Bibcode
3434:Bibcode
3387:Bibcode
3340:Bibcode
3161:Bibcode
3110:8262624
3033:Bibcode
2983:Bibcode
1714:is the
1237:is the
1068:, with
936:, with
817:removed
802:sources
744:photons
736:crystal
633:Leggett
608:Störmer
593:Bednorz
553:Giaever
523:Bardeen
513:Hubbard
488:Peierls
478:Onsager
428:Polymer
413:Colloid
376:Polaron
367:Plasmon
362:Exciton
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2091:Where
1537:Where
1204:Where
777:Theory
752:bosons
681:
648:Perdew
638:Parisi
598:MĂĽller
588:Rohrer
583:Binnig
573:Wilson
568:Fisher
528:Cooper
493:Landau
381:Magnon
357:Phonon
198:Plasma
98:Plasma
88:Liquid
53:Phases
3745:S2CID
3682:S2CID
3627:S2CID
3466:S2CID
3185:S2CID
3106:S2CID
3072:arXiv
2861:laser
2348:With
855:Near
548:Esaki
473:Bloch
468:Debye
463:Bragg
453:Onnes
386:Roton
83:Solid
3737:PMID
3729:ISSN
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3619:ISSN
3580:PMID
3562:ISSN
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3403:ISSN
3356:ISSN
3302:ISBN
3267:OCLC
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1800:and
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1000:the
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3219:doi
3169:doi
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2991:doi
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