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