Band in ARPES caused by photodissociation of Landau-Pekar polarons
Microsoft Word - statia.doc
Band in ARPES caused by photodissociation of Landau-Pekar polarons
A.E.Myasnikova, E.N. Myasnikov
South Federal University, 344090 Rostov-on-Don, Russia
We consider decay of phonon condensate into phonons at photodissociation of the strong
coupling large polaron (SCLP), when the charge carrier becomes free. Expression to calculate
the band in ARPES caused by photodissociation of SCLP is obtained. The band in ARPES
caused by photodissociation of strong-coupling large-radius polarons is a broad band with the
shape determined by Poisson distribution. It can be structured or unstructured depending on
the phonon dispersion since a distance between neighbouring lines comprising the band is the
phonon energy. Half-width of the band is in the interval 1.3 - 1.7Ep, depending on the phonon
energy. The band maximum is situated approximately at the electron energy Ephot-W-3.2Ep
(where Ephot is the photon energy, W is work function), and its position does not depend on
the electron wave vector direction.
PACS numbers 71.38.Fp, 74.25.Gz, 79.60.Bm
1. Introduction
We have shown earlier [1] that the strong-coupling large radius polaron (i.e. polarons
that occur at strong electron-phonon interaction in a medium where the conduction band is not
narrow) contains phonon condensate. (In other words, the polarization field in such polarons
is in quantum-coherent state.) If the charge carrier is removed from the SCLP on its
photodissociation the phonon condensate decays into phonons. As any Bose-condensate, the
phonon condensate is a superposition of states with different number of quanta where
summands are phased up to small quantum fluctuations. Therefore photodissociation of such
polarons is accompanied by radiation of different number of phonons in each act. Decay of
the phonon condensate at the SCLP photodissociation results in a wide band in the optical
conductivity spectrum. The calculated band in the optical conductivity spectrum [2] has the
maximum at about 4.2Ep and half-width about 2.5 Ep, where Ep is the polaron binding energy.
The calculated band is in good conformity with mid-IR band in the optical conductivity
spectra of complex oxides. The predicted ratio (≈4) of the maximum position of the band
caused by the SCLP photoionization to the frequency of maximum of the band caused by
phototransitions into polaron excited states calculated by other group [3] is also in good
conformity with experiments.
Obviously, decay of phonon condensate at the SCLP photodissociation should display
itself as sufficiently wide band in photoemission too. The aim of the present study is to obtain
an expression to calculate the band in ARPES spectrum caused by photodissociation of SCLP.
2. Photodissociation of the strong-coupling large polaron
Following Emin [] we calculate the band in the optical conductivity spectrum caused
by photodissociation of SCLP using the simplest treatment of the photoelectric effect.
Therefore our result can be easily reconstructed in order to describe the band in ARPES
caused by the SCLP photodissociation. So, the initial state of the system is a charge carrier in
autolocalized state  (coupled in the polarization potential well that represents a polarization
field in coherent state.
The polaron photodissociation occurs as a result of interaction of an electromagnetic
wave of frequency Ω  with the charge carrier in the polaron (the longitudinal field of the
polarization in the polaron obviously does not interact with the transverse electromagnetic
wave). The operator of the interaction has the form
       ( ) rQkA ie
H *intˆ
=  ,                                       (1)
where kh  is the electron momentum, A is the amplitude of the vector potential of the
electromagnetic field, related with its intensity I as it follows: cI hπ2/2AΩ= ; Q is the wave
vector of the electromagnetic wave. According to Fermi golden rule a probability of transition
of the system from the state i  into the state f  per unit time under the influence of the
operator intĤ  has the form
( )fiif EEiHfW −= δ
,      (2)
where iE  and  fE  are the energies of the initial and final states of the whole system. If the
initial state is the ground state [1,2] of the polaron in an electromagnetic field of a frequency
Ω  then
( ) ( )∏−+=
qdrri ββπβ exp17/
3        (3)
and iE = Ω+−− hWE p  where Ep is the polaron binding energy, W is work function, and
Pekar wave function for the carrier state in the polaron is used.
To describe the final state we following Emin [3] use the simplest treatment of the
photoelectric effect where the final state of the charge carrier is approximated as a free-carrier
state. The vectors of possible final states of the phonon field are the eigen vectors
{ } ∏=
qq νν  of the non-shifted Hamiltonian ∑ +=
qqq bbH ph ωhˆ
 describing the states
with the certain number of quanta qν  in each harmonics. Thus, after the photodissociation the
state (3) transforms into a state
( ) { }∏−=
qkr νiLf exp2
,      (4)
provided the sum of qν  (taking values 0 or 1) from the set { }qν  yields a certain number ν .
Hence, the energy of the final state is ωνh
, if we neglect the dependence of ω
on q. Thus,
( ) ⎟⎟
−−−Ω+−=− ωνδδ h
WEEE pfi
.      (5)
As the operator intĤ  acts only on the electron variables, the matrix element of the
transition has the form
 ( ) ( )∫ ∏−
qqkrr derHiLdiHf
r νβπβ β17/ˆexpˆ 3int2
.     (6)
Naturally, it contains the scalar product of the vector of a coherent state of the phonon field by
a vector of its state with the certain number of phonons. After carrying out the integration in
(18) the probability of the electron transition into a state with the wave vector with modulus
k  and direction in a spatial angle ϕθθ ddsin  has the form
( ) ( ) ( )∏⋅
qqk kkQ
ν ddLcm
3*, 17
where
( ) ( )
Ω== d
2 hh π
ρ k      (8)
is the spectral density of the final carrier states with the wave vector directed in the body
angle dΩ [3]. According to (5) the electron momentum kh  and energy ε  in the final state are
related as follows:
( ) ( )ωνεε hhh −−Ω== pEmmk ** 22 .       (9)
According to the energy and the momentum conservation laws (Exp.(5) and
0qkQ += , where q0 is the wave vector of the phonon field after the polaron photoionization,
qqq ν0 ) an experiment can measure only the probability (7) summarized over all
possible sets { }qν  having the same values of ∑=
qνν  and q0:
( ) ( )
( )kkQkA
qqk ρνββ
ν ddLcm
dW ∑∏⋅
3*, 17
,(10)
Here symbol ν  over Σ  denotes that the sum is carried out over the sets{ }qν  satisfying the
condition νν =∑
q . Besides, there is not a set with 0=ν  among the sets { }qν . Indeed, in
such a case 00 =q , hence, Qk = , and 0== QAkA , i.e. the probability of a transition with
appearance of such a set { }qν  is zero.
 Summarized probability (10) contains as a multiplier the sum
dP .                                (11)
The sum (11) calculated in [2] with taking into account the fact that dk<<1 has the form:
==∑∏ edP )!1(
qq                                (12)
where ν  is the average number of phonons radiated at the SCLP photodissociation [2]:
( ) 12 −= ων hpE .                                         (13)
To write the scalar product in (10) let us use ordinary for ARPES experiment
geometry [4] where the wave vector Q of the incident photon lies in XZ plane of the
coordinate system and makes angle ψ with z axes. The wave vector k of the electron is
considered to have a component k  lying in the XY plane of the coordinate system (and
coinciding with the sample surface) and ⊥k  in the perpendicular direction. Then
ψεψϕ sin/2coscos)( 22* AkmAk −+= hkA                 (14)
where the relation 22* /2 kmk −=⊥ hε   is taken into account. Finally, the wave vector Q
of the photon ordinarily used in ARPES experiments can be approximated as zero [].
Thus, expression (10) takes the form
( ) ( )
kAkmk
εψεψϕ
βπ )!1()1(
sin/2coscos
256 1
                                                                                                                                                                          (15)
Exp.(15) represents the probability of the polaron photodisociation with appearance of ν
phonons and electron with the energy ε and wave vector k in the body angle dΩ around
certain direction. The direction is determined by k projection on X-, Y- and Z-axes k cosφ,
k sinφ, and 22* /2 kmk −=⊥ hε , respectively. In Exp.(15) k(ε) is determined by Exp.(9).
Figs.1,2 demonstrate calculated by Exp.15 band in ARPES caused by
photodissociation of SCLP with the binding energy 0.14 eV and 0.17 eV, respectively. The
upper curves on both figures correspond to kx=1, ky=0, the lower curves correspond to kx=0,
ky=0. For both figures the electron-phonon interaction constant α=6 (it determines the phonon
energy for given Ep), W−Ωh =20eV and phonon dispersion is neglected. If do not neglect
the phonon dispersion the points on Figs.1,2 will transform into “partial” bands, and the
resulting summarized band can be structured or unstructured depending on the phonon
dispersion since a distance between neighbouring lines comprising the band is the phonon
energy. Half-width of the band is in the interval 1.3 - 1.7Ep, depending on the phonon energy.
The band maximum is situated approximately at the electron energy Ephot-W-3.2Ep (where
Ephot is the photon energy, W is work function), and its position does not depend on the
electron wave vector direction.
   0.5
     1
      1.5
     2
      2.5
19.3 19.4 19.5 19.6 19.7 19.8
g, eV
Fig.1. Band in ARPES caused by photoionization of SCLP with Ep=0.14 eV in neglect of
phonon dispersion, at α=6 and hΩ-W=20 eV. The upper and  lower “curves” correspond
to kx=1 and kx=0, respectively, ky=0.
     2
     4
     6
     8
19.2 19.3 19.4 19.5 19.6 19.7
Fig.2. Band in ARPES caused by photoionization of SCLP with Ep=0.17 eV in neglect of
phonon dispersion, at α=6 and hΩ-W=20 eV. The upper and  lower “curves” correspond
to kx=1 and kx=0, respectively, ky=0.
It is worth noting that intensity of the so-called “coherent” zero-phonon line (band) is equal to
zero as it can be seen from Figs1,2. The reason of this is explained below Exp.(10).
For example, we can calculate the polaron binding energies in YBa2Cu3O6+y,
Nd2CuO4-y, La2CuO4+y and La2-xSrxCuO4+y from the position of maximum of the mid-infrared
band in their optical conductivity spectra [5]. Then the maximum of the bands in ARPES of
these substances caused by photodissociation of the polarons will be approximately at the
electron energy (with respect to the energy Ephot-W)  -0.48eV, -0.52eV, -0.44eV and -0.4eV,
respectively.
Table 1
condoptΩh ,eV
Ep , eV ARPES
maxΩh , eV, with
respect to Ephot-W
Yba2Cu3O6+y 0.62±0.05 ≈0.155 ≈0.48
Nd2CuO4-y 0.76±0.01 ≈0.17 ≈0.52
La2CuO4+y 0.6±0.02 ≈0.14 ≈0.44
La2-xSrxCuO4+y 0.53±0.05 ≈0.13 ≈0.4
To make comparison of the theoretical results with experiments Figs.3,4 demonstrate
 ARPES spectra of underdoped La2-xSrxCuO4 [6] and Nd2-xCexCuO4 [7], respectively.
             
Fig.3. ARPES spectra of underdoped                 Fig.4. ARPES spectra of underdoped
La2-xSrxCuO4 [6]                                 Nd2-xCexCuO4 [7]
3. Temperature behavior of the band caused by SCLP photoionization
Velocity of SCLP is limited by the minimum phase (or maximum group) velocity u of
phonons participating in SCLP formation. (It is shown for phonon dispersion of the form
2 ku+= ωω  in [8]). As a result temperature of SCLP thermal destruction is much
lower than their binding energy Ep [9]. The temperature corresponding to double lowering of
the polaron concentration due to their thermal destruction can be approximated as  [10]
     
176.0
278.0 ⎟⎟
ET pc
 where m* is the bare carrier effective mass, u is the maximum group (and minimum phase)
velocity of phonons interacting with the charge carrier,  ∞−= εε /1/1 0c , 0ε  is a static
dielectric constant, ∞ε  is a dielectric permittivity at high frequencies, p0 - «maximum»
momentum of the carrier in the polaron [9].
An example of temperature dependence of SCLP concentration is demonstrated by
Fig.5 [10].
Fig.5. Temperature dependence of polaron concentration (curve 1), concentration of free
carriers with the momentums p<p0 (curve 2) and concentration of free carriers with the
momentums p>p0 (curve 3) in a system with total carrier concentration n=1.3n0, polaron
binding energy Ep=0.11eV, u=5*104cm/s. n0 is the maximum polaron concentration,
00 )2/()3/4(*2 hππpn = .
Accordingly, integral intensity of the band caused by SCLP photodissociation will
decrease as it occurs, e.g., in optical conductrivity spectra of 5233.0 OVNa−β  shown by
Fig.6 [11].
     Fig.6. Optical conductivity spectra of 5233.0 OVNa−β [11].
References
1. E. N. Myasnikov, A.E. Myasnikova, and F. V. Kusmartsev, Phys. Rev. B 72, 224303 (2005).
2. E. N. Myasnikov, A.E. Myasnikova, Z.P. Mastropas, cond-mat/0703693.
3. D. Emin, Phys. Rev.  B 48, 1369 (1993).
4. A. Damascelli, Z. Hussain and Z.X.Shen, Rev. Mod. Phys. 75, 473 (2003).
5. G. A. Thomas et al., Phys. Rev. B 45, 2474 (1992).
6. A. Ino et al., Phys. Rev. B 62, 4137 (2000).
7. N.P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002).
8. A. E. Myasnikova. Phys. Rev. B 52, 10457 (1995).
9.  E.  N. Myasnikov and A. E. Myasnikova, JETP 89, 746 (1999).
10.  A. E. Myasnikova. Phys. Let. A 291, 439 (2001).
11. C. Presura et al., Phys. Rev. Lett. 90, 026402, (2003).
