band in arpes caused by photodissociation of strong ...corpes07/beitraege/myasnikova.pdf ·...
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Band in ARPES caused byphotodissociation of strong-coupling
large polaronsA.E.Myasnikova, E.N.Myasnikov
South Federal University, Rostov-on-Don,Russia
We consider decay of phonon condensate into phonons atphotodissociation of the strong-coupling large polarons whenthe charge carrier becomes free. Expression to calculate the
band in ARPES caused by photodissociation of SCLP is obtained.
• Strong-coupling large polaron (SCLP)• Phonon condensate in SCLP• Decay of phonon condensate at photodissociation of SCLP• Band in optical conductivity and in ARPES due to
photodissociation of SCLP and its comparison with experiments
ElectronElectron-phononphonon interaction:interaction: FrohlichFrohlichHamiltonianHamiltonian
– Large radius (R>a) polaron occurs if the freecarrier band is larger than the polaron bindingenergy Ep. Continual approximation
– Strong electron-phonon coupling occurs if Ep>>(phonon energy). Adiabatic approximation
– As a result - strong-coupling large polaron(SCLP)
[ ]∑≠
−++
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+⎟⎠⎞
⎜⎝⎛−+∇−=
0
21
*2
*
2 2||2 k
ii ebebV
ebbm
Hr
hh
h krk
krk
kkkkr k ε
ωπω
∞
∞
−=
εεεε
ε0
0*
ωh
Phonon condensate in SCLPPhonon condensate in SCLP
Strong electron-phonon interaction shifts equilibriumpositions of the phonon field harmonics:
If is eigenfunction of Hamiltonian of non-shiftedoscillator than is eigenfunction of shiftedHamiltonian where
In case of adiabatic condition applicability a trial wavefunction can be written in the form of product:
∏=k
kUU kkkkk bdbdU *exp −= +
( ) ( ) 0exp,0
*0
⎭⎬⎫
⎩⎨⎧
−−= ∑≠
+
kkkkkRr bdbds βψ
nnU
*, kkkk dbdb == +
Finding parameters of phonon condensateFinding parameters of phonon condensatewith variational methodwith variational method
To determine we find minimum of
over and . It occurs at
,
(E.N.Myasnikov et al., Phys. Rev. B 72, 224303 (2005)).
In the minimum
The value of β is determined by minimization ofover β.
kkk
ϕiedd =
( )βηωεπ
kk
k k h*2
Ved = ( )kRkRk Cπϕ 2+−=
2*
2
2 r∇−−m
sHs hkd kϕ
sHs∑≠
−=∇−−0
*2*
2
2 kkkkr dd
msHs ωhh
PhotodissociationPhotodissociation ofof strongstrong-couplingcoupling polaronpolaron
For the polarization field the photodissocaition of SCLP is quickFranck-Condon process:
It cannot be represented as successive radiation of single phonons byelectron caused by electron-phonon interaction as it is done by
A.S. Mishchenko et al., Phys. Rev. Lett. 91, 236401 (2003), 96,
136405 (2006) since it occurs without participation of electron andelectron-phonon interaction.
( )fiif EEiHfW −= δπ 2int
2h
( ) rQkA iecm
eH *inth
=
( ) ( )∏−+=q
qdrri ββπβ exp17/3
( )∏−=
qqkr νiLf exp2
3
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−Ω+−=− ωνδδ h
hh *
22
2mEEE pfi
k∑ +=
qqqq bbH ph ωhˆ
∏∝2
qqif dW ν
Probability of the electron transition into a statewith the wave vector modulus k and directionin a body angle dΩ
( ) ( ) ( )∏⎭⎬⎫
⎩⎨⎧
−+=−−
qqq kkQkA ρνβ
βππ dd
LcmedW
22
32232/3* 1
7322 h
h
( )( )
Ω= dkLmd23
3*
2)(
hπερ k
( ) ( )ωνεε hhh −−−Ω== pEWmmk ** 22
∑=
v
vq
dWdWexp ∑=q
qνν
0qkQ += ∑=q
qqq ν0
Decay of phonon condensate
∑∏=ν
νν ν
q qqq
2dP
( ) 2212exp! k
kkk
kkk
k ddd −= ∏∏ − ννν
∑∞
=
=1
2
k
kkkkkν
ννω dE h
pEddEE 2* === ∑∑k
kkkk
k ωh
∑ ===k
k νωω hh /2/2pEEd average number of phonons
, radiated at SCLP photoionization
Taking into account that <<1 the calcultaion [E. N. Myasnikov, A.E.Myasnikova, Z.P. Mastropas, cond-mat/0703693.] yields
kd
νν
ν νν −
−
−= eP
)!1(
1
Optical conductivity
Mid-infrared band in the optical conductivity spectrum of Nd2CuO4-y :
experimental, at T=10K (diamonds) and calculated with α=6, pE =0.18 eV(solid curve).
2
),(Re
E
WN p
∞
∑ ΩΩ=
ε
νσ ν
h
0
0.005
0.01
0.015
0.02
0.025
0.03
0.2 0.4 0.6 0.8 1 1.2hS , eV
pE2.4max =Ωh
pE8.22.2 ÷=∆Ωh
νν
εεε
σ PEEm
Ne
pp
p ∑−
∞ ⎥⎥⎦
⎤
⎢⎢⎣
⎡Ω+
⎥⎥⎦
⎤
⎢⎢⎣
⎡Ω
Ω=
62/3
*
2 44.0)(144.0)(21
1024Re
ωνε hh −−Ω= pE
Mid-IR bands in complex oxides(.G. A. Thomas et al., Phys. Rev. B 45, 2474 (1992)).
ARPES from SCLPphotoionization
Band maximum atBand half-width
ψεψϕ sin/2coscos)( 22* AkmAk −+= hkA
22* /2 kmk −=⊥ hε
( ) ( )Ω
−⋅
+= −
−
− dek
kcmedW ν
ν
νν
βε
βπ )!1()1(7256 1
622
2
32*
2 kAh
( ) εε *2mk =h ωνε hh −−−Ω= pEW
pmax 3.2E-W-Ω≈Ω hh
p1.7E 1.3÷=∆Ωh
0
2
4
6
8
19.2 19.3 19.4 19.5 19.6 19.7g,eV
ARPES spectra caused by photoionization of SCLP in neglect of phonondispersion, α=6, hΩ-W=20 eV. The upper and lower “curves” correspond tokx=1 and kx=0, respectively, ky=0.
0
0.5
1
1.5
2
2.5
19.3 19.4 19.5 19.6 19.7 19.8g, eV
eVE p 14.0= eVEp 17.0=
photoionizΩh ,eV[14]
Ep , eV ARPESmaxΩh , eV, with
respect to Ephot-WernalintΩh ,
eV [14]YBa2Cu3O6+y 0.62±0.05 ≈0.155 ≈0.48 0.16±0.03Nd2CuO4-y 0.76±0.01 ≈0.17 ≈0.52 0.162±0.005La2CuO4+y 0.6±0.02 ≈0.14 ≈0.44 0.13±0.02La2-xSrxCuO4+y 0.53±0.05 ≈0.13 ≈0.4 0.16±0.03
Comparison with the experiment
ARPES spectra of underdoped La2-xSrxCuO4 andNd2-xCexCuO4 , respectively.
(A. Ino et al., Phys. Rev. B 62, 4137 (2000))
(N.P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002))
Temperature behavior of the band causedby SCLP photodissociation
Temperature of SCLP destruction is much lower than their bindingenergy Ep. It can be approximated as
where m* is the bare carrier effective mass, u is the maximumgroup (and minimum phase) velocity of phonons interacting withthe charge carrier ( ), , -«maximum» momentum of the carrier in the polaron.
176.0
0
*
278.0 ⎟⎟⎠
⎞⎜⎜⎝
⎛=
pucmET pc
∞−= εε /1/1 0c 0p2220
2 ku+= ωω
Accordingly, integral intensityof the band caused by SCLPphotodissociation willdecrease as it occurs inoptical conductrivityspectra of(C. Presura et al., Phys. Rev. Lett.
90, 026402, (2003).)
5233.0 OVNa−β