|
|
| Line 1: |
Line 1: |
| {{Condensed matter physics|expanded=Quasiparticles}}
| |
| A '''polaron''' is a [[quasiparticle]] used in [[condensed matter physics]] used to understand the interactions between [[electrons]] and [[atoms]] in a solid material. The polaron concept was first proposed by [[Lev Landau]] in 1933 to describe an electron moving in a [[dielectric]] [[crystal]] where the [[ions|atom]] move from their equilibrium positions to effectively screen the charge of an electron, known as a [[phonon]] cloud. This lowers the [[electron mobility]] and increases the electron's [[effective mass (solid-state physics)|effective mass]].
| |
|
| |
|
| The general concept of a polaron has been extended to describe other interactions between the electrons and ions in metals that result in a [[bound state]], or a lowering of energy compared to the non-interacting system. Major theoretical work has focused on solving Fröhlich and Holstein [[hamiltonian (quantum mechanics)|Hamiltonians]]. This is still an active field of research to find exact numerical solutions to the case of one or two electrons in a large [[crystal structure|crystal lattice]], and to study the case of many interacting electrons.
| |
|
| |
|
| Experimentally, polarons are important to the understanding of a wide variety of materials. The electron mobility in [[semiconductors|semiconductor]] can be greatly decreased by the formation of polarons. [[Organic semiconductors]] are also sensitive to polaronic effects, and is particularly relevant in the design of [[organic solar cells]] that effectively transport charge. The electron phonon interactions that form [[cooper pairs]] in [[type-I superconductors]] can also be modelled as a polaron, and two opposite [[spin (physics)|spin]] electrons may form a bipolaron sharing a phonon cloud. This has been suggested as a mechanism for [[cooper pair]] formation in [[type-II superconductors]]. Polarons are also important for interpreting the [[optical conductivity]] of these types of materials.
| | Poker är den mest uppskattad cirka itu kortspelet i världen samt har varit betagande folk därpå 1800-talet likväl extremt populära dessa dagar fullkomligt enkelt därför att det jätte- mer banal glädje till. Poker själv är ej en , saken där innehåller mängd skilda online casino video spel flitigt använder standard 5 card poker labb omdöme.<br><br>gav mig icke pro ett utvärdering. Dom skänker mej alldeles fria pengar att effektuera det varenda 7 dagar, såsom utför all såsom inneha club-kort. Detta vart baserat kungen uppsyn personliga mening villig Sands On line casino Resort.<br><br>När nya casino s öppnar Las Vegas, förbereda sig se ett fullständigt andel kändisar. lokala nyheterna i Las Vegas ropar att beskåda vem det som dom tillåts lokal i Las Vegas. Paris Hilton, Matt Damon, Brad skulle all vara förut ett ny kurs Casino.<br><br>Underben visar är att det ett bygel som hoppas kommer äntligen fresta dig åt undertecknande opp de där. webbplatser erbjuder samt inga bonusar såsom medför att kan utföra för helt avgiftsfri kapital som förtöja ditt konto. Armé erbjuds åt nya för att bevilja dom Checka spel webbplatsen förut sig själva, ändock det likaså böj.<br><br>Programvaran nedladdningar ni väljer nya video game programvaran hämtas . Dom ger 24 timmars telefon hjälp och chatt bistånd med programvaran. OnlineVegas på rutt casino erbjuder mer än hundra lockton, vilket bland mest bruten alla online-kasinon.<br><br>Det här borde vara logotyp som antingen är känt pro sin fantastiska casino video lockton alternativt ett online casino erbjuder tjänster bruten hög kvalitet. bliva en Jackpotjoy affiliate? faktorn kräver allmänt därför att fråga allena ifall affiliateprogram baksida av underben ni kommer att erhålla från planen. Den etta aspekten av någon casino 2014 affiliate flat befinner sig att justera vilken bildfil villig streck casino . Om du uppträder existera någon affiliate ett välrenommerade webbplats, enär du någon förbättring möjlighet att erhålla mycket fler mandat och innerligt mer pengar stäv allena.<br><br>Ett annan element som krävs att kunna villkoren stäv webbplatsen. Det det anses viktigt att personer bestå medvetna om strategierna hane vinner kungen online kasinon. Ifall du känd metoderna blir det flyktig pro dej att korrigera nedanför spel. pro nya individerna befinner sig det viktigt att dom första lite veta att segrar casino online. Inom denna genre har det preliminära skall deponeras tidsfrist vilken ni kommer att ringa framgångsrika beloppet.<br><br>En annan casino 2014 erbjuder blackjack-turneringar befinner sig BetUs. Deras prispott 5 lax dollar minimum och avgiften 10 dollar. Ett extra option är engelska hamnen, vilket skänker par blackjack turneringar i 7 dagar. Andra valmöjlighet Online Vegas Gaming Club. [http://Www.Guardian.Co.uk/search?q=Allihopa+lirare Allihopa lirare] startar likadan stadga samt tillåts 24 timmar en besegrare tillkännages. Dessa turneringar avsevärt mer såsom slot turneringar såsom kontrast till skrivbord turneringar.<br><br>icke mirakel ett lång tidrymd allmän pokerspel spark off, i synnerhet villig dom mindre begränsningar, du kommer att påträffa befinner sig faktiskt jätte- bäst säljande on-line hobbyspelare som genomför odla dåligt, att kan bra att föreställa kontanter att tänka ! Besökstrafik underbart samt någon del av Everest gemenskapens betyder det delar dess artister tillsammans dem.<br><br>Det finns några bruten belöning rundor på som skänker lirare möjligheten att segra jätte- mer. Ett av bonusar kommer att tredubbla spelare vinna samt andra ett sekundär lockton heter Honey . nya casinobonus är ett videospelautomat avdelning såsom tjugo fem kurs 5 hjuls slant slots. Temat pro denna sport är ett Alpine picknick. I denna förmån det föremål är att erhålla Extra kånka att avancera inom träd ackumulera odla flera honung krukor han kan. Fastän befinner sig trick denna bonus att undvika arga bin, ifall bina svärm om björnen han faller det icke genomförbart att ackumulera allihopa innerligt mer honung krukor. innebära att symboler används befinner sig bin, bikupor, picknick, skunkar, parkanläggning rangers naturligtvis björnar.<br><br>Denna första nya casinobonus kan rensas ett kalender år bruten 90 dagar Det ger nya casino belöningen går åt $600. Bonusen betalas ut i steg tio andel 25 bruten insättning $20, bundenhet gällande vilket är mindre. nAprak Tilt Poker: Aprak Tilt poker befinner sig berusad - laddade poker webbsida därborta kan få saken där ultimata pokerbonusen gällande webben.<br><br>Sannolikheter är när en on-line casino överlever att långa vanligaste syftet att dessa kasinon icke enastående samt fingerfärdig nog förut användning från ins outs från företaget. Ju längre har kasino stannat företaget tekniken har använda möjliga frågor såsom de flesta kasinon vissa att råka. Någon annan anledning varför flertal har osäkerheter nya casino webbplatser därför meriter. Ju längre online casino inneha varit i företaget desto befinner sig dess tillförlitlighet som det varenda klass att stoppa över i branschen inom många år.<br><br>For those who have virtually any questions about wherever and how you can utilize Vi listar samtliga senaste ultimata nya casino 2014!! [[http://peoplehelpingpeople.guru/members/latonz81/activity/75654/ gå til følgende område]], you are able to email us in our web site. |
| | |
| The polaron, a [[fermionic]] [[quasiparticle]], should not be confused with the [[polariton]], a [[bosonic]] [[quasiparticle]] analogous to a hybridized state between a photon and an optical phonon.
| |
| | |
| == Polaron theory==
| |
| | |
| [[L. D. Landau]] <ref name="Landau1933">
| |
| {{cite journal
| |
| | author = Landau LD | |
| | title = Über die Bewegung der Elektronen in Kristalgitter | |
| | journal = Phys. Z. Sowjetunion
| |
| | volume = 3
| |
| | pages = 644–645
| |
| | year = 1933
| |
| }}
| |
| </ref> and S. I. Pekar <ref name="Pekar1951">
| |
| {{cite journal
| |
| | author = Pekar SI
| |
| | title = Issledovanija po Elektronnoj Teorii Kristallov
| |
| | journal = Gostekhizdat, Moskva
| |
| | volume =
| |
| | pages =
| |
| | year = 1951
| |
| }}
| |
| </ref> formed the basis of polaron theory. A charge placed in a polarizable medium will be screened. [[Dielectric]] theory describes the phenomenon by the induction of a polarization around the charge carrier. The induced polarization will follow the charge carrier when it is moving through the medium. The carrier together with the induced polarization is considered as one entity, which is called a polaron (see Fig. 1).
| |
| | |
| [[Image:Polaron scheme1.svg|thumb|left|300px|Fig. 1: Artist view of a polaron.<ref name="Devreese1979">
| |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Moles agitat mentem. Ontwikkelingen in de fysica van de vaste stof
| |
| | journal = Rede uitgesproken bij de aanvaarding van het ambt van buitengewoon hoogleraar in de fysica van de vaste stof, in het bijzonder de theorie van de vaste stof, bij de afdeling der technische natuurkunde aan de Technische Hogeschool Eindhoven
| |
| | volume =
| |
| | pages = | |
| | year = 1979
| |
| }}
| |
| </ref> A conduction electron in an ionic crystal or a polar semiconductor repels the negative ions and attracts the positive ions. A self-induced potential arises, which acts back on the electron and modifies its physical properties.]]
| |
| | |
| {| class="wikitable" style="margin: 1em auto"
| |
| |+ Table 1: Fröhlich coupling constants <ref name="Devreese2005">
| |
| {{cite book
| |
| | last = Devreese |first=Jozef T. |authorlink=Jozef T. Devreese
| |
| | chapter = Polarons
| |
| | title = Encyclopedia of Physics (Third Edition) |editor1-first=R.G. |editor1-last=Lerner |editor2-first=G.L. |editor2-last=Trigg |publisher=Wiley-VCH |location=Weinheim
| |
| | volume = 2 | |
| | pages = 2004–2027 | |
| | year = 2005
| |
| | oclc = 475139057 | |
| }}
| |
| </ref>
| |
| ! Material
| |
| ! α
| |
| ! Material
| |
| ! α
| |
| |-
| |
| | InSb
| |
| | 0.023
| |
| | KI
| |
| | 2.5
| |
| |-
| |
| | InAs
| |
| | 0.052
| |
| | TlBr
| |
| | 2.55
| |
| |-
| |
| | GaAs
| |
| | 0.068
| |
| | KBr
| |
| | 3.05
| |
| |-
| |
| | GaP
| |
| | 0.20
| |
| | RbI
| |
| | 3.16
| |
| |-
| |
| | CdTe
| |
| | 0.29
| |
| | Bi<sub>12</sub>SiO<sub>20</sub>
| |
| | 3.18
| |
| |-
| |
| | ZnSe
| |
| | 0.43
| |
| | CdF<sub>2</sub>
| |
| | 3.2
| |
| |-
| |
| | CdS
| |
| | 0.53
| |
| | KCl
| |
| | 3.44
| |
| |-
| |
| | AgBr
| |
| | 1.53
| |
| | CsI
| |
| | 3.67
| |
| |-
| |
| | AgCl
| |
| | 1.84
| |
| | SrTiO<sub>3</sub>
| |
| | 3.77
| |
| |-
| |
| | α-Al<sub>2</sub>O<sub>3</sub>
| |
| | 2.40
| |
| | RbCl
| |
| | 3.81
| |
| |-
| |
| |}
| |
| {{clr}}
| |
| | |
| A conduction electron in an ionic crystal or a polar semiconductor is the prototype of a polaron. [[Herbert Fröhlich]] proposed a model [[Hamiltonian (quantum mechanics)|Hamiltonian]] for this polaron through which its dynamics are treated quantum mechanically (Fröhlich Hamiltonian).<ref name="Fröhlich1950">
| |
| {{cite journal
| |
| | author = [[Herbert Fröhlich|Fröhlich H]], Pelzer H, [[Sigurd Zienau|Zienau S]]
| |
| | title = | |
| | journal = Phil. Mag. | |
| | volume = 41 | |
| | pages = 221
| |
| | year = 1950
| |
| }}</ref><ref name="Fröhlich1954">
| |
| {{cite journal
| |
| | author = [[Herbert Fröhlich|Fröhlich H]]
| |
| | title = Electrons in lattice fields
| |
| | journal = Adv. Phys.
| |
| | volume = 3
| |
| | issue = 11
| |
| | pages = 325
| |
| | year = 1954
| |
| | doi = 10.1080/00018735400101213
| |
| |bibcode = 1954AdPhy...3..325F }}</ref>
| |
| This model assumes that electron wavefunction is spread out over many ions which are all somewhat displaced from their equilibrium positions, or the continuum approximation. The strength of the electron-phonon interaction is expressed by a dimensionless coupling constant α introduced by Fröhlich.<ref name="Fröhlich1954"/> In Table 1 the Fröhlich coupling constant is given for a few solids. The Fröhlich Hamiltonian for a single electron in a crystal using [[second quantization]] notation is:
| |
| <math>
| |
| H = H_{e} + H_{ph} + H_{e-ph}
| |
| </math> | |
| | |
| <math> | |
| H_{e} = \sum_{k,s} \xi(k,s) c_{k,s}^{\dagger} c_{k,s}
| |
| </math>
| |
| | |
| <math>
| |
| H_{ph} = \sum_{q,v} \omega_{q,s} a_{q,v}^{\dagger} a_{q,v} | |
| </math>
| |
| | |
| <math>
| |
| H_{e-ph} = \frac{1}{\sqrt{2N} } \sum_{k,s,q,v} \gamma(\alpha , q , k , v ) \omega_{qv} ( c_{k ,s}^{\dagger} c_{k-q , s} a_{q,v} + c_{k-q ,s}^{\dagger} c_{k , s} a^{\dagger}_{q,v} )
| |
| </math>
| |
| | |
| The exact form of gamma depends on the material and the type of phonon being used in the model. A detailed advanced discussion of the variations of the Fröhlich Hamiltonian can be found in J. T. Devreese and A. S. Alexandrov <ref name="Devreese2009">
| |
| {{cite journal
| |
| | author = J. T. Devreese and A. S. Alexandrov | |
| | title = Fröhlich polaron and bipolaron: recent developments
| |
| | journal = Rep. Prog. Phys.
| |
| | volume = 72
| |
| | pages = 066501
| |
| | year = 2009
| |
| | doi = 10.1088/0034-4885/72/6/066501
| |
| | arxiv = 0904.3682
| |
| | issue = 6
| |
| |bibcode = 2009RPPh...72f6501D }}
| |
| </ref> The terms Fröhlich polaron and large polaron are sometimes used synonymously, since the Fröhlich Hamiltonian includes the continuum approximation and long range forces. There is no known exact solution for the Fröhlich Hamiltonian with longitudinal optical (LO) [[phonons]] and linear <math> \gamma </math> (the most commonly considered variant of the Fröhlich polaron) despite extensive investigations.<ref name="Pekar1951" /><ref name="Devreese2005" /><ref name="Fröhlich1950" /><ref name="Fröhlich1954" /><ref name="Kuper1963">
| |
| {{cite journal
| |
| | author = Kuper GC, Whitfield GD (eds.)
| |
| | title = Polarons and Excitons | |
| | journal = Oliver and Boyd, Edinburgh | |
| | volume =
| |
| | pages =
| |
| | year = 1963
| |
| }}
| |
| </ref><ref name="Appel1968">
| |
| {{cite journal
| |
| | author = Appel J
| |
| | title =
| |
| | journal = In: Solid State Physics, F. Seitz, D. Turnbull, and H. Ehrenreich (eds.), Academic Press, New York
| |
| | volume = 21
| |
| | pages = pp.193–391
| |
| | year = 1968
| |
| }}
| |
| </ref><ref name="Devreese1972"> | |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]] (ed.)
| |
| | title = Polarons in Ionic Crystals and Polar Semiconductors
| |
| | journal = North-Holland, Amsterdam | |
| | volume = | |
| | pages = | |
| | year = 1972 | |
| }}
| |
| </ref><ref name="Mitra1987">
| |
| {{cite journal
| |
| | author = Mitra TK, Chatterjee A, Mukhopadhyay S
| |
| | title = Polarons
| |
| | journal = Phys. Rep.
| |
| | volume = 153 | |
| | issue = 2–3 | |
| | pages = 91 | |
| | year = 1987
| |
| | doi = 10.1016/0370-1573(87)90087-1
| |
| |bibcode = 1987PhR...153...91M }}
| |
| </ref><ref name="Devreese1996">
| |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]]
| |
| | title =
| |
| | journal = In" Encyclopedia of Applied Physics, G. L. Trigg (ed.), VCH, Weinheim | |
| | volume = 14 | |
| | pages = pp.383–413 | |
| | year = 1996
| |
| }}
| |
| </ref><ref name="Alexandrov1996"> | |
| {{cite journal
| |
| | author = Alexandrov AS, Mott N
| |
| | title = Polarons and Bipolarons
| |
| | journal = World Scientific, Singapore
| |
| | volume =
| |
| | pages =
| |
| | year = 1996
| |
| }}
| |
| </ref>
| |
| | |
| Despite the lack of an exact solution, some approximations of the polaron properties are known.
| |
| | |
| The physical properties of a polaron differ from those of a band-carrier. A polaron is characterized by its ''self-energy'' <math>\Delta E</math>, an ''effective mass'' <math>m*</math> and by its characteristic ''response'' to external electric and magnetic fields (e. g. dc mobility and optical absorption coefficient).
| |
| | |
| When the coupling is weak (<math>\alpha</math> small), the self-energy of the polaron can be approximated as:<ref name="Smondyrev1986">
| |
| {{cite journal
| |
| | author = Smondyrev MA
| |
| | title = Diagrams in the polaron model
| |
| | journal = Theor. Math. Phys.
| |
| | volume = 68
| |
| | issue = 1 | |
| | pages = 653
| |
| | year = 1986
| |
| | doi = 10.1007/BF01017794
| |
| |bibcode = 1986TMP....68..653S }}
| |
| </ref>
| |
| | |
| {|
| |
| |width=500|<math>
| |
| \frac{\Delta E}{\hbar\omega } \approx -\alpha -0.015919622\alpha^2,
| |
| </math>
| |
| |<math>(1)\,</math>
| |
| |}
| |
| | |
| and the polaron mass <math>m*</math>, which can be measured by cyclotron resonance experiments, is larger than the band mass m of the charge carrier without self-induced polarization:<ref name="Röseler1968">
| |
| {{cite journal
| |
| | author = Röseler J
| |
| | title = A new variational ansatz in the polaron theory
| |
| | journal = Physica Status Solidi (b)
| |
| | volume = 25
| |
| | issue = 1 | |
| | pages = 311 | |
| | year = 1968 | |
| | doi = 10.1002/pssb.19680250129
| |
| |bibcode = 1968PSSBR..25..311R }}
| |
| </ref>
| |
| | |
| {|
| |
| |width=500|<math>
| |
| \frac{m^*}{m} \approx 1+\frac{\alpha}{6}+0.0236\alpha^2.
| |
| </math>
| |
| |<math>(2)\,</math>
| |
| |}
| |
| | |
| When the coupling is strong (α large), a variational approach due to Landau and Pekar indicates that the self-energy is proportional to α² and the polaron mass scales as α⁴. The Landau-Pekar variational calculation
| |
| <ref name="Pekar1951"/>
| |
| yields an upper bound to the polaron self-energy <math>E < -C_{PL} \alpha^2 </math>, valid
| |
| for ''all'' α, where <math>C_{PL}</math> is a constant determined by solving an integro-differential equation. It was an open question for many years whether this
| |
| expression was asymptotically exact as α tends to infinity. Finally,
| |
| Donsker and Varadhan,<ref name="Donsker1983">M. Donsker and R.Varadhan(1983).
| |
| "Asymptotics for the Polaron", ''Commun. Pure Appl. Math.'' '''36''', 505–528.</ref> applying [[large deviation theory]] to Feynman's
| |
| path integral formulation for the self-energy, showed the large α exactitude
| |
| of this Landau-Pekar formula. Later, Lieb and Thomas
| |
| <ref name="Lieb1997">{{cite journal | author = Lieb E.H., Thomas L.E. | year = 1997 | title = Exact Ground State Energy of the Strong Coupling Polaron | url = | journal = Commun. Math. Physics | volume = 183 | issue = 3| pages = 511–519 | doi = 10.1007/s002200050040 |arxiv = cond-mat/9512112 |bibcode = 1997CMaPh.183..511L }}</ref> gave a shorter proof using more conventional methods,
| |
| and with explicit bounds on the lower order corrections to the
| |
| Landau-Pekar formula.
| |
| | |
| [[Feynman]] <ref name="Feynman1955">
| |
| {{cite journal
| |
| | author = [[Richard Feynman|Feynman RP]]
| |
| | title = Slow Electrons in a Polar Crystal
| |
| | journal = Phys. Rev.
| |
| | volume = 97
| |
| | issue = 3
| |
| | pages = 660
| |
| | year = 1955 | |
| | doi = 10.1103/PhysRev.97.660 | |
| |bibcode = 1955PhRv...97..660F }}
| |
| </ref> introduced a [[variational principle]] for path integrals to study the polaron. He simulated the interaction between the electron and the polarization modes by a harmonic interaction between a hypothetical particle and the electron. The analysis of an exactly solvable ("symmetrical") 1D-polaron model,<ref name="Devreese1964">
| |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]], Evrard R | |
| | title = On the excited states of a symmetrical polaron model
| |
| | journal = Phys. Lett.
| |
| | volume = 11
| |
| | issue = 4
| |
| | pages = 278
| |
| | year = 1964
| |
| | doi = 10.1016/0031-9163(64)90324-5
| |
| |bibcode = 1964PhL....11..278D }}
| |
| </ref><ref name="Devreese1968">
| |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]], Evrard R
| |
| | title =
| |
| | journal = Proceedings of the British Ceramic Society
| |
| | volume = 10
| |
| | pages = 151
| |
| | year = 1968
| |
| }}
| |
| </ref> Monte Carlo schemes <ref name="Mishchenko2000">
| |
| {{cite journal
| |
| | author = Mishchenko AS, Prokof'ev NV, Sakamoto A, Svistunov BV | |
| | title = Diagrammatic quantum Monte Carlo study of the Fröhlich polaron
| |
| | journal = Phys. Rev. B
| |
| | volume = 62
| |
| | issue = 10
| |
| | pages = 6317 | |
| | year = 2000 | |
| | doi = 10.1103/PhysRevB.62.6317
| |
| |bibcode = 2000PhRvB..62.6317M }}
| |
| </ref><ref name="Titantah2001"> | |
| {{cite journal
| |
| | author = Titantah JT, Pierleoni C, Ciuchi S
| |
| | title = Free Energy of the Fröhlich Polaron in Two and Three Dimensions
| |
| | journal = Phys. Rev. Lett. | |
| | volume = 87
| |
| | pages = 206406
| |
| | year = 2001
| |
| | doi = 10.1103/PhysRevLett.87.206406 | |
| | pmid = 11690499 | |
| | issue = 20 | |
| | bibcode=2001PhRvL..87t6406T
| |
| |arxiv = cond-mat/0010386 }}
| |
| </ref> and other numerical schemes <ref name="DeFilippis2003">
| |
| {{cite journal
| |
| | author = De Filippis G, Cataudella V, Marigliano Ramaglia V, Perroni CA, Bercioux D | |
| | title = Ground state features of the Fröhlich model | |
| | journal = Eur. Phys. J. B
| |
| | volume = 36 | |
| | issue = 1 | |
| | pages = 65
| |
| | year = 2003
| |
| | doi = 10.1140/epjb/e2003-00317-x
| |
| |arxiv = cond-mat/0309309 |bibcode = 2003EPJB...36...65D }}
| |
| </ref> demonstrate the remarkable accuracy of Feynman's path-integral approach to the polaron ground-state energy. Experimentally more directly accessible properties of the polaron, such as its mobility and optical absorption, have been investigated subsequently. | |
| | |
| == Polaron optical absorption ==
| |
| The expression for the magnetooptical absorption of a polaron is:<ref name="Peeters1986">
| |
| {{cite journal
| |
| | author = Peeters FM, [[Jozef T. Devreese|Devreese JTL]] | |
| | title = Magneto-optical absorption of polarons
| |
| | journal = Phys. Rev. B
| |
| | volume = 34
| |
| | issue = 10
| |
| | pages = 7246
| |
| | year = 1986
| |
| | doi = 10.1103/PhysRevB.34.7246
| |
| |bibcode = 1986PhRvB..34.7246P }}
| |
| </ref>
| |
| | |
| {|
| |
| |width=500|<math>
| |
| \Gamma(\Omega) \propto -\frac{\mathrm{Im} \Sigma(\Omega)}{\left[\Omega-\omega_{\mathrm{c}}-\mathrm{Re} \Sigma(\Omega)\right]^2 + \left[\mathrm{Im}\Sigma(\Omega)\right]^2} .
| |
| </math>
| |
| |<math>(3)\,</math>
| |
| |}
| |
| | |
| Here, <math>\omega_{c}</math> is the [[cyclotron frequency]] for a rigid-band electron. The magnetooptical absorption Γ(Ω) at the frequency Ω takes the form Σ(Ω) is the so-called "memory function", which describes the dynamics of the polaron. Σ(Ω) depends also on α, β<sub> what is beta?</sub> and <math>\omega_{c}</math>.
| |
| | |
| In the absence of an external magnetic field (<math>\omega_{c}=0</math>) the optical absorption spectrum (3) of the polaron at weak coupling is determined by the absorption of radiation energy, which is reemitted in the form of LO phonons. At larger coupling, <math>\alpha \ge 5.9</math>, the polaron can undergo transitions toward a relatively stable internal excited state called the "relaxed excited state" (RES) (see Fig. 2). The RES peak in the spectrum also has a phonon sideband, which is related to a Franck-Condon-type transition.
| |
| | |
| [[Image:FigP24-2.jpg|thumb|200px|left|Fig.2. Optical absorption of a polaron at <math>\alpha = 5</math> and 6. The RES peak is very intense compared with the Franck-Condon (FC) peak.<ref name="Devreese1972"/><ref name="Devreese1972b">
| |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]], De Sitter J, Goovaerts M
| |
| | title = Optical Absorption of Polarons in the Feynman-Hellwarth-Iddings-Platzman Approximation
| |
| | journal = Phys. Rev. B
| |
| | volume = 5
| |
| | issue = 6
| |
| | pages = 2367 | |
| | year = 1972
| |
| | doi = 10.1103/PhysRevB.5.2367
| |
| |bibcode = 1972PhRvB...5.2367D }}
| |
| </ref>]]
| |
| | |
| A comparison of the DSG results <ref name="Devreese1972b"/> with the [[optical conductivity]] spectra given by approximation-free numerical <ref name="Mishchenko2003">
| |
| {{cite journal
| |
| | author = Mishchenko AS, Nagaosa N, Prokof'ev NV, Sakamoto A, Svistunov BV
| |
| | title = Optical Conductivity of the Fröhlich Polaron | |
| | journal = Phys. Rev. Lett.
| |
| | volume = 91
| |
| | pages = 236401
| |
| | year = 2003
| |
| | doi = 10.1103/PhysRevLett.91.236401
| |
| | pmid = 14683203
| |
| | issue = 23
| |
| | bibcode=2003PhRvL..91w6401M
| |
| |arxiv = cond-mat/0312111 }}
| |
| </ref> and approximate analytical approaches is given in ref.<ref name="DeFilippis2006">
| |
| {{cite journal
| |
| | author = De Filippis G, Cataudella V, Mishchenko AS, Perroni CA, [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Validity of the Franck-Condon Principle in the Optical Spectroscopy: Optical Conductivity of the Fröhlich Polaron
| |
| | journal = Phys. Rev. Lett.
| |
| | volume = 96
| |
| | pages = 136405
| |
| | year = 2006
| |
| | doi = 10.1103/PhysRevLett.96.136405
| |
| | pmid = 16712012 | |
| | issue = 13 | |
| | bibcode=2006PhRvL..96m6405D
| |
| |arxiv = cond-mat/0603219 }}
| |
| </ref>
| |
| | |
| Calculations of the [[optical conductivity]] for the Fröhlich polaron performed within the Diagrammatic Quantum Monte Carlo method,<ref name="Mishchenko2003"/> see Fig. 3, fully confirm the results of the path-integral variational approach <ref name="Devreese1972b"/> at <math>\alpha \lesssim 3.</math> In the intermediate coupling regime <math>3<\alpha <6,</math> the low-energy behavior and the position of the maximum of the [[optical conductivity]] spectrum of ref.<ref name="Mishchenko2003"/> follow well the prediction of ref.<ref name="Devreese1972b"/> There are the following qualitative differences between the two approaches in the intermediate and strong coupling regime: in ref.,<ref name="Mishchenko2003"/> the dominant peak broadens and the second peak does not develop, giving instead rise to a flat shoulder in the [[optical conductivity]] spectrum at <math>\alpha =6</math>. This behavior can be attributed to the optical processes with participation of two <ref name="Goovaerts1973">
| |
| {{cite journal
| |
| | doi = 10.1103/PhysRevB.7.2639
| |
| | author = Goovaerts MJ, De Sitter J, [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Numerical Study of Two-Phonon Sidebands in the Optical Absorption of Free Polarons in the Strong-Coupling Limit| journal = Phys. Rev.
| |
| | volume = 7
| |
| | issue = 6
| |
| | pages = 2639
| |
| | year = 1973
| |
| |bibcode = 1973PhRvB...7.2639G }}
| |
| </ref> or more phonons. The nature of the excited states of a polaron needs further study.
| |
| | |
| [[Image:PolaronOptCond.jpg|thumb|right|200px|Fig. 3: Optical conductivity spectra calculated within the Diagrammatic Quantum Monte Carlo method (open circles) compared to the DSG calculations (solid lines).<ref name="Devreese1972b"/><ref name="Mishchenko2003"/>]]
| |
| | |
| The application of a sufficiently strong external magnetic field allows one to satisfy the resonance condition <math>\Omega =\omega _{\mathrm{c}}+\mathrm{Re} \Sigma (\Omega )</math>, which {(for <math>\omega_c < \omega</math>)} determines the polaron cyclotron resonance frequency. From this condition also the polaron cyclotron mass can be derived. Using the most accurate theoretical polaron models to evaluate <math>\Sigma (\Omega )</math>, the experimental cyclotron data can be well accounted for.
| |
| | |
| Evidence for the polaron character of charge carriers in AgBr and AgCl was obtained through high-precision cyclotron resonance experiments in external magnetic fields up to 16 T.<ref name="Hodby1987">
| |
| {{cite journal
| |
| | author = Hodby JW, Russell GP, Peeters F, [[Jozef T. Devreese|Devreese JTL]], Larsen DM
| |
| | title = Cyclotron resonance of polarons in the silver halides: AgBr and AgCl | |
| | journal = Phys. Rev. Lett.
| |
| | volume = 58
| |
| | pages = 1471–1474
| |
| | year = 1987
| |
| | doi = 10.1103/PhysRevLett.58.1471 | |
| | pmid = 10034445 | |
| | issue = 14 | |
| | bibcode=1987PhRvL..58.1471H
| |
| }}
| |
| </ref> The all-coupling magneto-absorption calculated in ref.,<ref name="Peeters1986"/> leads to the best quantitative agreement between theory and experiment for AgBr and AgCl. This quantitative interpretation of the cyclotron resonance experiment in AgBr and
| |
| AgCl <ref name="Hodby1987"/> by the theory of ref.<ref name="Peeters1986"/> provided one of the most convincing and clearest demonstrations of Fröhlich polaron features in solids.
| |
| | |
| Experimental data on the magnetopolaron effect, obtained using far-infrared photoconductivity techniques, have been applied to study the energy spectrum of shallow donors in polar semiconductor layers of CdTe.<ref name="Grynberg1996">
| |
| {{cite journal
| |
| | author = Grynberg M, Huant S, Martinez G, Kossut J, Wojtowicz T, Karczewski G, Shi JM, Peeters FM, [[Jozef T. Devreese|Devreese JTL]] | |
| | title =
| |
| | journal =
| |
| | volume =
| |
| | pages = | |
| | year = 1996
| |
| }}
| |
| </ref>
| |
| | |
| The polaron effect well above the LO phonon energy was studied through cyclotron resonance measurements, e. g., in II-VI semiconductors, observed in ultra-high magnetic fields.<ref name="Miura2003">
| |
| {{cite journal
| |
| | author = Miura N, Imanaka Y
| |
| | title = Polaron cyclotron resonance in II–VI compounds at high magnetic fields
| |
| | journal = Physica Status Solidi (b) | |
| | volume = 237
| |
| | issue = 1
| |
| | pages = 237
| |
| | year = 2003
| |
| | doi = 10.1002/pssb.200301781
| |
| |bibcode = 2003PSSBR.237..237M }}
| |
| </ref> The resonant polaron effect manifests itself when the cyclotron frequency approaches the LO phonon energy in sufficiently high magnetic fields.
| |
| | |
| == Polarons in two dimensions and in quasi-2D structures ==
| |
| | |
| The great interest in the study of the two-dimensional electron gas (2DEG) has also resulted in many investigations on the properties of polarons in two dimensions.<ref name="Devreese1987">
| |
| {{cite journal
| |
| | author = [[Jozef T. Devreese|Devreese JTL]], Peeters FM (eds.)
| |
| | title = The Physics of the Two-Dimensional Electron Gas
| |
| | journal = ASI Series, Plenum, New York
| |
| | volume = B157 | |
| | pages = | |
| | year = 1987 | |
| }}
| |
| </ref><ref name="Wu1986">
| |
| {{cite journal
| |
| | author = Wu XG, Peeters FM, [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Effect of screening on the optical absorption of a two-dimensional electron gas in GaAs-Al<sub>x</sub>Ga<sub>1-x</sub>As heterostructures | |
| | journal = Phys. Rev. B | |
| | volume = 34
| |
| | issue = 4
| |
| | pages = 2621
| |
| | year = 1986
| |
| | doi = 10.1103/PhysRevB.34.2621
| |
| |bibcode = 1986PhRvB..34.2621W }}
| |
| </ref><ref name="Peeters1987"> | |
| {{cite journal
| |
| | author = Peeters FM, [[Jozef T. Devreese|Devreese JTL]] | |
| | title = Scaling relations between the two- and three-dimensional polarons for static and dynamical properties | |
| | journal = Phys. Rev. B
| |
| | volume = 36 | |
| | issue = 8 | |
| | pages = 4442 | |
| | year = 1987 | |
| | doi = 10.1103/PhysRevB.36.4442 | |
| |bibcode = 1987PhRvB..36.4442P }}
| |
| </ref> A simple model for the 2D polaron system consists of an electron confined to a plane, interacting via the Fröhlich interaction with the LO phonons of a 3D surrounding medium. The self-energy and the mass of such a 2D polaron are no longer described by the expressions valid in 3D; for weak coupling they can be approximated as:<ref name="Sak1972">
| |
| {{cite journal
| |
| | author = Sak J | |
| | title = Theory of Surface Polarons | |
| | journal = Phys. Rev. B
| |
| | volume = 6 | |
| | issue = 10 | |
| | pages = 3981
| |
| | year = 1972
| |
| | doi = 10.1103/PhysRevB.6.3981
| |
| |bibcode = 1972PhRvB...6.3981S }}
| |
| </ref><ref name="Peeters1988">
| |
| {{cite journal
| |
| | author = Peeters FM, Wu XG, [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Exact and approximate results for the mass of a two-dimensional polaron | |
| | journal = Phys. Rev. B | |
| | volume = 37
| |
| | issue = 2
| |
| | pages = 933
| |
| | year = 1988 | |
| | doi = 10.1103/PhysRevB.37.933
| |
| |bibcode = 1988PhRvB..37..933P }}
| |
| </ref>
| |
| | |
| {|
| |
| |width=500|<math>
| |
| \frac{\Delta E}{\hbar \omega} \approx -\frac{\pi}{2}\alpha\ - 0.06397\alpha^2;
| |
| </math>
| |
| |<math>(4)\,</math>
| |
| |}
| |
| | |
| {|
| |
| |width=500|<math>
| |
| \frac{m^*}{m} \approx 1+\frac{\pi}{8}\alpha\ + 0.1272348\alpha^2.
| |
| </math>
| |
| |<math>(5)\,</math>
| |
| |}
| |
| | |
| It has been shown that simple scaling relations exist, connecting the physical properties of polarons in 2D with those in 3D. An example of such a scaling relation is:<ref name="Peeters1987"/>
| |
| | |
| {|
| |
| | width=500 |<math>
| |
| \frac{m^{*}_{2D}(\alpha)}{m_{2D}}=\frac{m^{*}_{3D}(\frac{3}{4}\pi\alpha)}{m_{3D}} </math>,
| |
| |<math>(6)\,</math>
| |
| |}
| |
| | |
| where <math>m_\mathrm{2D}^*</math> (<math>m_\mathrm{3D}^*</math>) and <math>m_\mathrm{2D}^{}</math> (<math>m_\mathrm{3D}^{}</math>) are, respectively, the polaron and the electron-band masses in 2D (3D).
| |
| | |
| The effect of the confinement of a Fröhlich polaron is to enhance the ''effective'' polaron coupling. However, many-particle effects tend to counterbalance this effect because of screening.<ref name="Devreese1987" /><ref name="DasSarma1985">
| |
| {{cite journal
| |
| | author = Das Sarma S, Mason BA
| |
| | title = Optical phonon interaction effects in layered semiconductor structures
| |
| | journal = Ann. Phys. (New York)
| |
| | volume = 163
| |
| | issue = 1 | |
| | pages = 78 | |
| | year = 1985 | |
| | doi = 10.1016/0003-4916(85)90351-3 | |
| }}
| |
| </ref>
| |
| | |
| Also in 2D systems [[cyclotron resonance]] is a convenient tool to study polaron effects. Although several other effects have to be taken into account (nonparabolicity of the electron bands, [[many-body]] effects, the nature of the confining potential, etc.), the polaron effect is clearly revealed in the cyclotron mass. An interesting 2D system consists of electrons on films of liquid He.<ref name="Shikin1973">
| |
| {{cite journal
| |
| | author = Shikin VB, Monarkha YP | |
| | title = | |
| | journal = Sov. Phys. – JETP
| |
| | volume = 38
| |
| | pages = 373
| |
| | year = 1973
| |
| }}
| |
| </ref><ref name="Jackson1981">
| |
| {{cite journal
| |
| | author = Jackson SA, Platzman PM
| |
| | title = Polaronic aspects of two-dimensional electrons on films of liquid He
| |
| | journal = Phys. Rev. B | |
| | volume = 24
| |
| | issue = 1
| |
| | pages = 499
| |
| | year = 1981
| |
| | doi = 10.1103/PhysRevB.24.499 | |
| |bibcode = 1981PhRvB..24..499J }}
| |
| </ref> In this system the electrons couple to the ripplons of the liquid He, forming "ripplopolarons". The effective coupling can be relatively large and, for some values of the parameters, self-trapping can result. The acoustic nature of the ripplon dispersion at long wavelengths is a key aspect of the trapping.
| |
| | |
| For GaAs/Al<sub>x</sub>Ga<sub>1-x</sub>As quantum wells and superlattices, the polaron effect is found to decrease the energy of the shallow donor states at low magnetic fields and leads to a resonant splitting of the energies at high magnetic fields. The energy spectra of such polaronic systems as shallow donors ("bound polarons"), e. g., the D<sub>0</sub> and D<sup>-</sup> centres, constitute the most complete and detailed polaron spectroscopy realised in the literature.<ref name="Shi1993">
| |
| {{cite journal
| |
| | author = Shi JM, Peeters FM, [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Magnetopolaron effect on shallow donor states in GaAs
| |
| | journal = Phys. Rev. B | |
| | volume = 48 | |
| | issue = 8 | |
| | pages = 5202 | |
| | year = 1993
| |
| | doi = 10.1103/PhysRevB.48.5202
| |
| |bibcode = 1993PhRvB..48.5202S }}
| |
| </ref>
| |
| | |
| In GaAs/AlAs quantum wells with sufficiently high electron density, anticrossing of the cyclotron-resonance spectra has been observed near the GaAs transverse optical (TO) phonon frequency rather than near the GaAs LO-phonon frequency.<ref name="x">
| |
| {{cite journal
| |
| | author = Poulter AJL, Zeman J, Maude DK, Potemski M, Martinez G, Riedel A, Hey R, Friedland KJ | |
| | title = Magneto Infrared Absorption in High Electron Density GaAs Quantum Wells | |
| | journal = Phys. Rev. Lett. | |
| | volume = 86
| |
| | pages = 336–9
| |
| | year = 2001 | |
| | doi = 10.1103/PhysRevLett.86.336
| |
| | pmid = 11177825
| |
| | issue = 2
| |
| | bibcode=2001PhRvL..86..336P
| |
| |arxiv = cond-mat/0012008 }}
| |
| </ref> This anticrossing near the TO-phonon frequency was explained in the framework of the polaron theory.<ref name="x">
| |
| {{cite journal
| |
| | author = Klimin SN, [[Jozef T. Devreese|Devreese JTL]] | |
| | title = Cyclotron resonance of an interacting polaron gas in a quantum well: Magnetoplasmon-phonon mixing | |
| | journal = Phys. Rev. B | |
| | volume = 68 | |
| | issue = 24 | |
| | pages = 245303 | |
| | year = 2003
| |
| | doi = 10.1103/PhysRevB.68.245303
| |
| |arxiv = cond-mat/0308553 |bibcode = 2003PhRvB..68x5303K }}
| |
| </ref> | |
| | |
| Besides optical properties,<ref name="Devreese2005"/><ref name="Devreese1996"/><ref name="Calvani2001">
| |
| {{cite journal
| |
| | author = Calvani P
| |
| | title = Optical Properties of Polarons
| |
| | journal = Editrice Compositori, Bologna
| |
| | volume =
| |
| | pages =
| |
| | year = 2001
| |
| }}
| |
| </ref> many other physical properties of polarons have been studied, including the possibility of self-trapping, polaron transport,<ref name="Feynman1962">
| |
| {{cite journal
| |
| | author = Feynman RP, Hellwarth RW, Iddings CK, Platzman PM
| |
| | title = Mobility of Slow Electrons in a Polar Crystal
| |
| | journal = Phys. Rev.
| |
| | volume = 127
| |
| | issue = 4
| |
| | pages = 1004
| |
| | year = 1962
| |
| | doi = 10.1103/PhysRev.127.1004
| |
| |bibcode = 1962PhRv..127.1004F }}
| |
| </ref> magnetophonon resonance, etc.
| |
| | |
| == Extensions of the polaron concept ==
| |
| | |
| Significant are also the extensions of the polaron concept: acoustic polaron, [[piezoelectric]] polaron, electronic polaron, bound polaron, trapped polaron, [[Spin (physics)|spin]] polaron, molecular polaron, solvated polarons, polaronic exciton, Jahn-Teller polaron, small polaron, [[bipolaron]]s and many-polaron systems.<ref name="Devreese2005"/> These extensions of the concept are invoked, e. g., to study the properties of conjugated polymers, colossal magnetoresistance perovskites, high-<math>T_{c}</math> superconductors, layered MgB<sub>2</sub> superconductors, fullerenes, quasi-1D conductors, semiconductor nanostructures.
| |
| | |
| The possibility that polarons and bipolarons play a role in high-<math>T_{c}</math> [[superconductors]] has renewed interest in the physical properties of many-polaron systems and, in particular, in their optical properties. Theoretical treatments have been extended from one-polaron to many-polaron systems.<ref name="Devreese2005"/><ref name="Bassani2003">
| |
| {{cite journal
| |
| | author = Bassani FG, Cataudella V, Chiofalo ML, De Filippis G, Iadonisi G, Perroni CA
| |
| | title = Electron gas with polaronic effects: beyond the mean-field theory
| |
| | journal = Physica Status Solidi (b)
| |
| | volume = 237
| |
| | issue = 1
| |
| | pages = 173
| |
| | year = 2003
| |
| | doi = 10.1002/pssb.200301763
| |
| |bibcode = 2003PSSBR.237..173B }}
| |
| </ref><ref name="Hohenadler2007">
| |
| {{cite journal
| |
| | author = Hohenadler M, Hager G, Wellein G, Fehske H
| |
| | title = Carrier-density effects in many-polaron systems
| |
| | journal = J. Phys.: Condens. Matter
| |
| | volume = 19
| |
| | issue = 25
| |
| | pages = 255210
| |
| | year = 2007
| |
| | doi = 10.1088/0953-8984/19/25/255210
| |
| |arxiv = cond-mat/0611586 |bibcode = 2007JPCM...19y5210H }}
| |
| </ref>
| |
| | |
| A new aspect of the polaron concept has been investigated for semiconductor [[nanostructures]]: the exciton-phonon states are not factorizable into an adiabatic product Ansatz, so that a ''non-adiabatic'' treatment is needed.<ref name="Fomin1998">
| |
| {{cite journal
| |
| | author = Fomin VM, Gladilin VN, [[Jozef T. Devreese|Devreese JTL]], Pokatilov EP, Balaban SN, Klimin SN
| |
| | title = Photoluminescence of spherical quantum dots
| |
| | journal = Phys. Rev. B
| |
| | volume = 57
| |
| | issue = 4
| |
| | pages = 2415
| |
| | year = 1998
| |
| | doi = 10.1103/PhysRevB.57.2415
| |
| |bibcode = 1998PhRvB..57.2415F }}
| |
| </ref> The ''non-adiabaticity'' of the exciton-phonon systems leads to a strong enhancement of the phonon-assisted transition probabilities (as compared to those treated adiabatically) and to multiphonon optical spectra that are considerably different from the [[Franck-Condon]] progression even for small values of the electron-phonon coupling constant as is the case for typical semiconductor nanostructures.<ref name="Fomin1998"/>
| |
| | |
| In biophysics [[Davydov soliton]] is a propagating along the [[protein]] [[α-helix]] self-trapped amide I excitation that is a solution of the Davydov Hamiltonian. The mathematical techniques that are used to analyze Davydov's soliton are similar to some that have been developed in polaron theory. In this context the [[Davydov soliton]] corresponds to a ''polaron'' that is (i) ''large'' so the continuum limit approximation in justified, (ii) ''acoustic'' because the self-localization arises from interactions with acoustic modes of the lattice, and (iii) ''weakly coupled'' because the anharmonic energy is small compared with the phonon bandwidth.<ref name="Scott1992">
| |
| {{cite journal
| |
| | author = Scott AS
| |
| | title = Davydov's soliton
| |
| | journal = Physics Reports
| |
| | volume = 217
| |
| | issue = 1
| |
| | pages = 1–67
| |
| | year = 1992
| |
| | doi = 10.1016/0370-1573(92)90093-F
| |
| |bibcode = 1992PhR...217....1S }}
| |
| </ref>
| |
| | |
| More recently it was shown that the system of an impurity in a [[Bose-Einstein condensate]] is also a member of the polaron family.<ref name="Tempere2009">
| |
| {{cite journal
| |
| | author = Tempere J, Casteels W, Oberthaler M, Knoop S, Timmermans E and [[Jozef T. Devreese|Devreese JTL]]
| |
| | title = Feynman path-integral treatment of the BEC-impurity polaron
| |
| | journal = Phys. Rev. B
| |
| | volume = 80
| |
| | issue = 18
| |
| | pages = 184504
| |
| | year = 2009
| |
| | doi = 10.1103/PhysRevB.80.184504
| |
| |bibcode = 2009PhRvB..80r4504T |arxiv = 0906.4455 }}
| |
| </ref> This is very promising for experimentally probing the hitherto inaccessible strong coupling regime since in this case interaction strengths can be externally tuned through the use of a [[Feshbach resonance]].
| |
| | |
| ==See also==
| |
| | |
| * [[Sigurd Zienau]]
| |
| | |
| ==References and notes==
| |
| {{reflist|2}}
| |
| | |
| ==External links==
| |
| | |
| {{particles}}
| |
| | |
| [[Category:Quasiparticles]]
| |
| [[Category:Ions]]
| |