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Theory of photoelectron spectroscopy for organic molecules and their crystals Takashi Fujikawa ∗ , Kaori Niki, Hiroto Sakuma Graduate School of Advanced Integration of Science, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, Japan

a r t i c l e

i n f o

Article history: Available online 10 July 2015 Keywords: Photoemission Keldysh Green’s function Organic solids Electron–phonon interaction

a b s t r a c t In this short review we discuss recent progress in photoemission theory for organic molecules and their crystals. We discuss some important features in Keldysh Green’s function theory for the photoemission. We brieﬂy discuss many-body aspects in photoemission from core and extended levels. In particular phonon effects are investigated in more detail since organic solids are typically soft where electron–phonon interaction is important. Debye–Waller factor suppresses the interference effects of photoelectron waves which makes ARPES analyses useless, particularly in high energy region. © 2015 Elsevier B.V. All rights reserved.

1. Introduction A useful and practical many-body approach to the photoemission theory has been developed by Bardyszewski and Hedin [1] by use of projection operator techniques. Further reﬁnement is found in Ref. [2]. This theoretical approach can describe the interference between intrinsic (shake-up) and extrinsic losses for the loss band. The loss mechanism in which the photoelectron suffers inelastic loss during the travel on its way out through the surface is called “extrinsic”. These two loss mechanisms are not possible to separate sharply. Alternative ﬁrst principle many-body theory of photoemission processes can be built on Keldysh Green’s function theory as developed by Caroli et al. [3]. These theories have great advantages, for example, feasibility of temperature effects [3–5], radiation ﬁeld screening effects [6,7], and relativistic effects [8]. We have some good review articles on photoemission theory. Sébilleau [9] has given a nice introductory overview of photoemission theory which should be helpful both experimental and theoretical researchers. Schattke et al. [10] also have given an overview of core and valence photoemission. They discuss radiation ﬁeld effect which is rarely discussed, but is important. General many-body theory of core electron photoemission is discussed and reviewed by Hedin [11] based on his many-body scattering theory and quasi-boson models. This review is quite important to understand basic framework of the photoemission theory, but is rather difﬁcult to understand. Fujikawa [12] has written a review article on recent progress in photoemission theory, which also covers

relativistic effects and radiation ﬁeld screening. The article written by Almbladh and Hedin [13] is a landmark of photoemission theory, but the recent progress is not discussed there. For the studies of photoemission spectra from organic molecules and their crystals angle-resolved photoemission (ARPES) spectra are extensively used in UPS region which mainly provide us with information on band dispersion [14–17]. We however have some questions how phonon effects contribute to ARPES spectra. We can also wonder whether ARPES analyses work even in high energy region; it is well known that Gelius formula works so well in the XPS analyses (see Section 3.2). In this short review, we extensively discuss some important theoretical aspects of photoemission spectra, in particular from organic solids, on the basis of Keldysh Green’s function approach. Some topics, for example relativistic effects and so on, are not discussed here since they are not so important in photoemission spectra from organic systems composed of light elements.

2. Keldysh Green’s function theory For the later discussion we discuss some introductory materials. The question of choosing scattering states with incoming or outgoing spherical waves in photoemission has been discussed in detail within the one-electron theory [13]. Here we only stress that the photoemission state should be written for many-body systems

|n−∗ k = ∗ Corresponding author. Tel.: +81 43 290 3699; fax: +81 43 290 3699. E-mail address: [email protected] (T. Fujikawa). http://dx.doi.org/10.1016/j.elspec.2015.07.001 0368-2048/© 2015 Elsevier B.V. All rights reserved.

−i 1 † † a |n∗ = ak |n∗ + V |n∗ , E − H − i k E − H − i k †

†

Vk = [H, ak ] − εk ak , E = En∗ + εk , εk =

k2 , 2

(1)

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176 †

where ak is the creation operator for the photoelectron state with – boundary condition. We measure the photoelectron current with momentum k in the remote future leaving the system in the state † |n* . The ﬁrst term ak |n∗ describes the intrinsic (shake-up) processes, and the second term describes the extrinsic and other dynamical coupling between the photoelectron and the target [2]. The basic difﬁculty in the nonequilibrium ﬁeld theory is to keep track of which operators are negatively (+ leg; ∞→ − ∞) or positively (− leg; −∞ → ∞) time ordered, i.e., on what contour they lie. One way to keep track of this is to artiﬁcially distinguish the external perturbation V(−) which takes the system forward in time from V(+) which takes the system backward in time [18]. When an external time-dependent perturbation is applied, the nonequilibrium electron Green’s function G(1, 2), (1 = (x1 , t1 ), x1 = (r1 , 1 )) is written as Tc

iG(1, 2) =

[S (+) S (−)

ˆ (1) ˆ † (2)]

S (+) S (−)

S (+) = T˜ exp −i

(3)

†

dx1 U(1)(1), ˆ (1) =

(1) (1).

(4)

∂ − h(x1 ) − U(1) − VH (1) G(1, 2) ∂t1 ıG(1, 2) = ıc (1, 2). ıU(3)

(6)

We use one-electron operator h(x1 ) deﬁned as h(x1 ) = Te (x1 ) −

˛

=

ı(t1 − t2 ), t1 , t2 ∈ − (7)

= 0, otherwise.

d3(1, 3)G(3, 2) = ıc (1, 2),

vc (1, 2) = v(r1 − r2 )ıc (t1 , t2 ).

d3d4G(1, 3) (32; 4)W (4, 1+ ).

(1, 2) = i

(12)

c

The integro-differential equation (11) is converted to the integral equation (Dyson equation)

d3d4G0 (1, 3)(3, 4)G(4, 2).

(13)

The vertex function satisﬁes the equation ı(1, 2) . ıV (3)

(14)

The screened Coulomb interaction W(1, 2) is given by use of the inverse dielectric function ε−1

d3ε−1 (1, 3)vc (3, 2) = vc (1, 2) +

W (1, 2) = c

d3d4vc (1, 3) (3, 4)vc (4, 2),

(15)

c

ε(1, 2) = ıc (1, 2) −

(16)

(17)

d3vc (1, 3)P(3, 2),

ı(1) P(1, 2) = = −i ıV (2)

d3d4G(1, 3) (34, 2)G(4, 1+ )

(19)

+

d2vc (1, 2)G(2, 2 ). c

(9)

(20)

This is what we call GW approximation [19]. Beyond the GW approximation, vertex correction (the second term of Eq. (14)) should be included. Depending on the time legs which t1 and t2 are on, we have four different Green’s functions. Their spectral representations for the ground state without time-dependent external perturbation are given by g c (x1 , x2 ; ω) =

fq (x1 )fq∗ (x2 ) ω − εq + i

fq (x1 )fq∗ (x2 )

(8)

d2vc (1, 2)(2) = −i

(18)

c

(12, 3) ≈ ıc (1, 2)ıc (1, 3),

g˜ (x1 , x2 ; ω) = −

The Hartree potential VH (1) is given by

(11)

c

The electron selfenergy is now given by the conventional form [19] except for the time integral on the closed path

q

The related bare Coulomb interaction vc is deﬁned as

c

(1, 2) = iG(1, 2)W (2, 1+ ).

= −ı(t1 − t2 ), t1 , t2 ∈ +,

VH (1) =

∂ − h(x1 ) − U(1) − VH (1) G(1, 2) − ∂t1

we obtain the lowest order approximation

Z˛ . |r1 − R˛ |

The ﬁrst term describes the kinetic energy, and the second term describes the Coulomb interaction between an electron at r1 and nuclei at R˛ with nuclear charge Z˛ . We have introduced delta function ıc on the closed contour C, which is deﬁned ıc (t1 − t2 )

i

The reducible and irreducible polarization propagators and P are important to describe the polarization due to dynamical electron–electron interaction. These coupled equations (Hedin’s equation) provide us with systematic approximations starting from the approximation in Eq. (14)

c

The symbol t1 ∈ − means that t1 is on − leg. From the equation of motion and the above relations, we obtain

c

we thus obtain the equation on closed path C,

(5)

ıS (+) = −iT˜ [S (+) (1)], ˆ (t1 ∈ +). ıU(1)

d3vc (1, 3)

(10)

ı(1) = (1) − (1),

ıS (−) = −iT [S (−) (1)], ˆ (t1 ∈ −), ıU(1)

−i

V (1) = U(1) + VH (1).

i (1, 2) = Tc [ı(1)ı(2)],

We can prove that [19]

c

ıG−1 (1, 2) ıV (4) , ıV (4) ıU(3)

(12, 3) = ıc (1, 2)ıc (1, 3) +

i

d4

c

where Tc , T˜ and T are path-ordering, anti time-ordering and timeordering operators. We use interaction representation of V as Vˆ . We now consider an external perturbation

G(1, 2) = G0 (1, 2) +

−∞

Vˆ (1) =

ıG−1 (1, 2) = ıU(3)

dt Vˆ (+) (t) , dt Vˆ (−) (t) ,

S (−) = T exp −i

With aid of a chain rule,

(2)

−∞

∞∞

,

169

q

ω − εq − i

g > (x1 , x2 ; ω) = −2 i g < (x1 , x2 ; ω) = 2 i

q

+

gn (x1 )g ∗ (x2 ) n

n

−

ω − εn − i

gn (x1 )g ∗ (x2 ) n

n

ω − εn + i

fq (x1 )fq∗ (x2 )ı(ω − εq ),

gn (x1 )gn∗ (x2 )ı(ω − εn ).

n

( → +0), , (21)

170

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176

The Dyson orbitals for the hole gn and particle propagation fq are deﬁned as [19] gn (x) = n∗ | (x)|0, fq (x) = 0| (x)|q, N + 1,

(22)

εn = E0 (N) − En (N − 1), εq = Eq (N + 1) − E0 (N).

In the Hartree–Fock approximation, the Dyson orbitals gn and fq correspond to the occupied and vacant orbitals whose orbital energies are εn and εq . We should note that the hole Dyson orbitals gn are localized on the target. Some of the particle Dyson orbitals fq are localized, whereas photoelectron Dyson orbital can propagate to the detector far from the target. In addition to these four Green’s functions, other two Green’s functions, retarded gr and advanced ga Green’s functions, are also important for the photoemission analyses g r (x1 , x2 ; ω) =

fq (x1 )fq∗ (x2 ) q

g a (x1 , x2 ; ω) =

ω − εq + i

fq (x1 )fq∗ (x2 ) q

ω − εq − i

+

gn (x1 )g ∗ (x2 ) n

n

+

ω − εn + i

gn (x1 )g ∗ (x2 ) n

n

ω − εn − i

,

√ which asymptotically approaches to the plane wave exp(ik · r)/ V . We can show that it is a solution of the one-electron equation [21]

dx a (x, x ; εk )fk− (x ) = εk fk− (x)

(24)

for the non-hermitian energy dependent “optical potential” a , which describes the ﬁnite mean free path of photoelectrons inside solids [22]. The screened Coulomb interaction W is quite important to describe loss and resonance effects observed in photoemission processes. Explicit spectral representation for the Keldysh components are shown for the stationary states W > (x1 , x2 ; ω) = −2 i

(28)

where We is just the electron screened Coulomb interaction given by (15). The density–density correlation function for the nuclei D is introduced here iD(1, 2) = Tc [ın (1)ın (2)],

(29)

ın (1) = n (1) − n (1).

If we neglect the effect of the lattice vibrations and put D = 0, the screened Coulomb interaction reduces to that obtained for a rigid lattice. 3. Photoemission theory

− fk− (1) = 0| (1)|0k

W (1, 2) = We (1, 2) + [We DW e ](1, 2),

(23) .

The lesser Green’s function g< has only localized components which show exponential decay far from the target. In contrast other ﬁve Green’s functions have scattering components fk− which signiﬁcantly contribute to the photoelectron current there. The photoelectron Dyson orbital fk− is deﬁned as given by Eq. (22)

h(x)fk− (x) +

Inelastic scattering processes during photoelectron propagation is described by the ﬂuctuation potential [2,4]. So far we have discussed electrons in a ﬁxed nuclear conﬁguration. Now nuclear motion is explicitly taken into account. Quantum ﬁeld theory has been developed to calculate phonon effects on the electron selfenergy by Baym [20], Hedin and Lundqvist [19]. They show that the screened Coulomb interaction W should be replaced by

v∗n (x1 )vn (x2 )ı(ω − ωn ),

In the ﬁeld theory to describe the photoelectron current excited from many-electron systems, the current density induced by X-ray ﬁeld is given at a detection position r and time t j(r, t) ∝

∂

∂r

−

∂ ∂r

gA< (xt, x t)|x=x ,

(30)

where we should take derivative before putting x = x , and use x = (r, ). In order to describe the nonequilibrium processes, we should calculate the Green’s function gA< including all interactions which can be calculated by use of Keldysh Green’s function GA (1, 2). That satisﬁes the Dyson equation

i

∂ − h(1) − A(1) − VH (1) GA (1, 2) − ∂t1

d3A (1, 3)GA (3, 2) = ıc (1, 2).

(31)

c

In the above equation c means the time integration on the Keldysh contour from −∞ to ∞ (− leg) and back to −∞ (+ leg). The interaction between the probe X-ray and an electron A is written as A(1) = a(t1 ) (r1 ) + c.c.

(32)

which describe the extrinsic losses. Here we have deﬁned the Fourier transform as

Conventional X-ray sources give harmonic time-dependence as exp(− iωt). X-ray free electron laser gives speciﬁc pulse-like timedependence for a(t). Both GA and A include the inﬂuence from the time-dependent electron–photon interaction. Now we deﬁne the correction ı caused by the X-ray probe: ı = A − . We thus can expand GA in terms of G without the electron–photon interaction, also ı and A,

W > (x1 , x2 ; ω) =

GA = G + G(ı + A)G + G(ı + A)G(ı + A)G + . . .

n>0

W < (x1 , x2 ; ω) = −2 i

vn (x1 )v∗n (x2 )ı(ω + ωn ),

(25)

n>0

dtW > (x1 , x2 ; t) exp(iωt).

˜ describe the resonance photoemiOn the other hand Wc and W ssion, which include the bare Coulomb interaction

˜ p (x1 , x2 ; ω) ˜ (x1 , x2 ; ω) = −v(r1 − r2 ) + W W

(26)

˜ p depend on ω. The ﬂuctuwhere the polarization parts Wpc and W ation potential vn (x) (n = / 0) is deﬁned by

vn (x) =

v(r − r )n|(x )|0dx .

The ﬁrst and the second order corrections to due to the interaction A are then given by use of the GW approximation (20) ı(1) (1, 2) = i(GAG)(1, 2)W (2, 1+ ),

W c (x1 , x2 ; ω) = v(r1 − r2 ) + Wpc (x1 , x2 ; ω)

(27)

(33)

(2)

ı

+

(1, 2) = i(GAGAG)(1, 2)W (2, 1 ).

(34) (35)

We measure stationary photoelectron current which should be caused by 2nd order processes in regard to A [4]. We pick up all possible terms in the order of A2 and W in the expansion of GA : (2)

GA = GAGAG + GAGı(1) G + Gı(1) GAG + Gı(2) G.

(36)

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176 (2)

Full expression of GA is now given by use of the time integration on the closed path c

(2)

GA (1, 2) =

(GXG)< = g < X a g a + g r X < g a + g r X r g < .

(38)

∞

d3d4g < (1, 3)X a (3, 4)g a (4, 2).

(g < X a g a )(1, 2) = −∞

We measure the photoelectron current at far from the target. The lesser Green’s function g< (xt, 3) shows the exponential decay as r→ ∞, so that we ﬁnd that

d3d4g < (xt, 3)X a (3, 4)g a (4, x t) → 0. On the other hand, both gr and ga have the contribution from the scattering states as shown by Eq. (23) which arrive at a detector [4]. Therefore only the second term in Eq. (38) has ﬁnite contribution. Substitution of the asymptotic formulas into Eq. (30) yields the following formula since ∇ fk− (r) = ikfk− (r) at r→ ∞

d1

d2fk− (1)∗ X < (1, 2)fk− (2).

(39)

The photoelectron current has to be real, which gives rise to the photoemission intensity Ik measuring the photoelectron momentum k

d1

d2fk− (1)∗ X < (1, 2)fk− (2).

(40)

After we give explicit formulas for X< , we will discuss the photoemission processes from now on.

3.1. Lowest order approximation; core excitation

ﬁrst term, the screened Coulomb interaction W is included which contributes to the extrinsic losses due to W> and W< , and the res˜ (anticausal). First we focus onance effects due to Wc (causal) and W on the simplest term, the ﬁrst term in Eq. (36), where X< is now given by (41)

The lesser Green’s function g< (1, 2) has information on the initial bound states from which photoelectrons are excited. For the conventional X-ray sources, we use a well known formula for the electron–photon interaction, A(1) ∝ (x1 ) exp(−iωt1 ).

∝

|fk− | |gn |2 ı(εk − ω − εn ),

n

(42)

k2 . 2

This formula is important to describe typical photoemission processes. From a dressed one-electron state gn , an electron is excited to the photoelectron state fk− by an X-ray photon, which suffers elastic scatterings from surrounding atoms and decays inside the solid under the inﬂuence of the non-Hermitian nonlocal selfenergy (optical potential) a (εk ). We ﬁrst focus on the core excitation: As usual we use an approximation (x) ≈ c (x)b where c is the core function and b is the associated annihilation operator from which an electron is excited; it is insensitive to molecular or solid electronic states. Now we study main photoemission band, n = 0, and satellite bands (n = / 0). The Dyson orbital gn (x) is thus given by gn (x) ≈ c (x)Sn , Sn = n∗ |b|0,

(43)

where Sn is intrinsic (shake-up) loss amplitude, and S0 is quite close to 1. For the practical calculations of the amplitude fk− | | c , one can use Bloch functions to approximate the photoelectron wave function fk− [24]. We rather prefer to using real space expression to extend the applicability even to disordered systems. When the potential v is given as the sum of non overlapping atomic potentials

v=

v˛ ,

˛

we have an expression for the total T-matrix expanded in terms of the site T-matrix t˛ and damping free propagator g0 (ε) = (ε − Te + i )

−1

( > 0),

where is the imaginary part of the optical potential a in the outer region of each atomic sphere, T=

All of the four terms in Eq. (36) are in the order of A2 : Except the

X < (1, 2) = A(1)g < (1, 2)A(2).

Ik ∝ Imfk− | g < (εk − ω) ∗ |fk−

εn = E0 (N) − En (N − 1), εk =

For example the full expression of the ﬁrst term is now given by use of the ordinary integration from −∞ to ∞

Ik ∝ Im

(37)

c

Next we project out the lesser components of the terms in the above formula. Langreth theorem converts the time integration on c to the ordinary time integration from −∞ to ∞ [23]. Let us consider an expression containing a generic operator X that will be speciﬁed in the following:

j(r, t) ∝ −ik

Substituting this expression and the spectral representation for g< shown in Eq. (21), we obtain useful formulas for the photoemission intensity

d3d4d5[G(1, 3)A(3)G(3, 4)ı(1) (4, 5)G(5, 2) + . . .] + . . .

d3d4[G(1, 3)A(3)G(3, 4)A(4)G(4, 2)] + c

171

˛

t˛ +

tˇ g0 t˛ + . . ., (44)

ˇ= / ˛

t˛ = v˛ + v˛ g0 t˛ . By use of these multiple scattering expansions, we obtain a practical formula for the photoemission amplitude − fk− | | c = Ak | | c +

˛( = / A)

k0 |t˛ gA | c +

− gA = g0 + g0 tA g0 , k0 |(1 + tA g0 ) = Ak |,

k0 |tˇ g0 t˛ gA | c + . . .

ˇ= / ˛( = / A)

(45)

To go one step further, we use electric dipole approximation for the electron–photon interaction. For the linear polarization parallel to the z-axis, the operator is given as ∝ z ∝ rY 10 (ˆr).

172

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176

For this polarization and mufﬁn-tin approximation, we obtain the amplitude by use of the multiple scattering expansion (45), fk− | | c =

exp(−ik · R˛ )

LL

˛

ˆ × (1 − X)−1 YL (k)

(1 − X)−1 = 1 + X + X 2 + X 3 + . . .,

˛A L L

(z) = 0 exp

MLLc , (46)

˛ˇ

XL L = (1 − ı˛ˇ )tl˛ (k)GL L (kR˛ˇ ), where R˛ is the position vector of scatterer ˛ measured from the photoelectron emitter A. Free Green’s function in angular momentum representation GL L (kR˛ˇ ) describes electron propagation from the site ˇ with angular momentum L = (l, m) to the site ˛ with angular momentum L = (l , m ). The matrix X is labelled by a set of atomic sites (A, ˛, ˇ, . . .), and angular momenta L and L . Full multiple scattering is taken into account by the inverse matrix (1 − X)−1 . The dipole matrix MLLc excited by the linearly polarized light ( z) is given by [25–27]

MLLc =

2 −l i exp(iıAl (k))(l)c G(Lc 10 | L)

(47)

where (l)c is a radial dipole integral between the lth partial wave of the photoelectron wave function Rl (kr) and the core function Rlc (r)

(l)c =

Rl (kr)Rlc (r)r 3 dr.

The Gaunt integral

G(L10 | L ) =

atoms, we expect that the DDF (z) should show a simple exponential law

dˆrYL (ˆr)Y10 (ˆr)YL∗ (ˆr)

has important selection rules, m = m, l = l ± 1. When we use an approximation (1 − X)−1 ≈ 1 + X we call single scattering approximation. In general, in the high-energy region, the scatterings from the surrounding atoms are quite weak so that low order (single, double scatterings, etc.) are sufﬁcient to explain the observed XPS spectra. In contrast the largest orbital angular momentum to be considered lmax is proportional to the photoelectron wave vector length k. We thus expect that electrons are dominantly scattered in the forward direction. As an other important feature of the high energy scattering the phase shifts ıl ’s are insensitive to the details of the electronic structures of the scattering atoms. Furthermore fast photoelectrons are mainly scattered in the forward direction, which is called forward focusing effect. By use of these speciﬁc features we can use high-energy photoemission as a tool to study surface and near surface structures, and is called X-ray photoelectron diffraction (XPD) [28]:XPD analyses provide us with direct structural information such as molecular orientation by use of the focusing effects. Besides the focusing effect, some interference can affect XPD patterns. Kazama et al. [29] have calculated angular distributions for photoelectrons from O 1s and Ge 1s levels of H2 O and GeCl2 to discuss the relative importance of the forward focusing and double-slit effects in XPD spectra. In the low- and intermediateenergy region, the double-slit effect can be observed. In contrast only forward focusing peaks are given in high-energy region. The double-slit effects can be observed under special conditions where the scattering amplitudes from different atoms have ﬁnite overlap in the bisecting direction. On the other hand, in the low energy region, the scatterings from near neighbors are considerably strong, so that the renormalized full multiple scatterings are crucial. Shinotsuka et al. have applied the multiple-scattering formula (46) to discuss depth distribution function [30]. If we neglect elastic scatterings from composite

−z

(48)

where z is the depth from the surface, is the inelastic mean free path (IMFP). Their XPD calculations, however, show some prominent structures due to the forward focusing effects. Except for near surface region, the DDF including elastic scattering follows the simple exponential decay shown by Eq. (48). We thus can obtain the IMFP from the analyses of the DDF, but the IMFP has inﬂuence of the elastic scatterings; less than 10% difference is introduced. 3.2. Lowest order approximation; valence excitation Discussion on photoemission from delocalized states is also based on Eq. (42): we should note that the delocalized states are still localized inside the target but are extended over it. We thus apply Eq. (42). Here we consider only the main band associated with the excitation from , gn (x) ≈ (x)S0n , S0n ≈ 1. In order to describe the photoemission from organic molecules and their crystals, the delocalized one-electron state can be assumed to be written as linear combination of atomic orbitals ˛ localized on the site ˛

(r) =

c˛ ˛ (r − R˛ ).

(49)

˛

For the moment we neglect spin effects. The amplitude fk− | | is thus written fk− | | = =

c˛ fk− | |˛

˛ − c˛ [ ˛k | |˛ +

ˇ( = / ˛)

˛

k0 |tˇ g˛ |˛ +

k0 |t g0 tˇ g˛ |˛ + . . .].

= / ˇ( = / ˛)

(50) − The photoelectron function excited from ˛th atom ˛k | is deﬁned similar to that in Eq. (45). The ﬁrst term in the large parenthesis describes the direct photoemission amplitude without suffering elastic scatterings from neighbors. The second term describes the single elastic scatterings, and so on. In the UPS region it is sufﬁcient to choose lmax from 3 to 5 to ensure convergence in regard to the calculated spectra. However the multiple scattering renormalization is important. In the mufﬁntin potential approximation for the linear polarization parallel to z axis, the amplitude is similar to Eq. (46) [25],

fk− | | =

˛ˇ

c˛ exp(−ik · Rˇ˛ )

ˆ × (1 − X)−1 YL (k)

ˇ˛ L L

MLL˛ .

LL

(51) In UPS region lmax is only from 3 to 5 so that the full multiple scattering calculations are practical even for quite large clusters. Shang et al. [31] have applied the above multiple scattering formula to the angular dependence of UPS spectra excited from the highest occupied molecular orbitals of NiPc and CoPc (Pc = phthalocyanine), which are in plane orbitals dominated by C 2p atomic orbitals in Pc ligands. A satisfactory agreement with the experimental angular distribution is obtained. Pushnig et al. suggested a new approach to gain information on the spatial electron distribution in molecular orbitals based on a simple approximation. Only the ﬁrst term in Eq. (50) is taken

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176

into account [32,33], which can be considerably simpliﬁed if initial molecular orbital is comprised of atomic orbitals ˛ with the same character. A speciﬁc example is a molecular orbital of planar polyatomic molecule containing only carbon and hydrogen atoms like pentacene. Then the coefﬁcients c˛ are only non-zero for pz orbitals, and the amplitude is approximated by

fk− | | ≈ N10 (k) N10 (k) =

c˛ exp(−ik · R˛ ),

˛

(52)

ˆ ML,10 YL (k).

L

The term N10 (k) does not depend on the atomic positions, but ˆ and εk . We should note that ML,10 depends on k depends on k as shown by Eq. (47). If the term N10 (k) is weakly varying funcˆ the photoemission angular dependence is dominated tion of k, by the term ˛ c˛ exp(− ik · R˛ ), which reﬂects the initial molecular orbital structure. They successfully applied this idea to the analyses of angle-resolved UPS spectra of pentacene [33]. We however have some questions about the reliability of the lowest order approximation where all scatterings from nearby atoms are completely neglected even in the UPS region: typically multiple scattering renormalization is supposed to be important there. For the analyses of X-ray absorption near edge structure (XANES) spectra (εk < 50 eV), full renormalized multiple scatterings are inevitable [34]. Recent works on photoemission also demonstrates the importance of multiple scattering renormalization in those low energy region [30]. Dauth et al. point out the importance of the Dyson orbital gn in Eq. (42) for the initial states. The imaging experiments do not show molecular orbitals, but Dyson orbitals [36]. They check whether Dyson-orbital and molecular-orbital based interpretation of ARPES lead to differences that are relevant on the experimentally observable scale. They conclude that a detailed understanding of ARPES experiments for organic solids requires one to go beyond the molecular orbital point of view. A simple theoretical model has been proposed to study electron–phonon interaction in ARPES spectra [35]. A sound theoretical basis is built on Keldysh theory [5]. We can rewrite the ﬁrst expression of Eq. (42) in terms of the retarded Green’s function gr instead of the lesser Green’s function g< Ik ∝ −2f (εk − ω)Imfk− | g r (εk − ω) ∗ |fk−

(53)

where f(ω) is the Fermi distribution function. In the Keldysh Green’s function theory, the retarded Green’s function satisﬁes a closed Dyson equation g r (ω) = g0r (ω) + g0r (ω)r (ω)g r (ω). Keeping diagonal elements of the selfenergy r in regard to band states , we obtain a convenient formula for UPS analyses Ik ∝ f (εk − ω) × |r (ω)|

|f − | | |2 k

2

(εk − ω − ε − ) + 2 ,

(54)

= ı ( − i ), ( > 0).

From ARPES analyses we obtain useful information about ε + and . They provide us with useful information on the electron–phonon interaction mediated selfenergy ph . Prominent band bending is observed near Fermi level caused by electron–phonon interaction [35]. Hatch et al. investigate the electron–phonon interaction in pentacene ﬁlms grown on Bi(0 0 1) using UPS spectra, which reveal thermal broadening [37]. They determine an electron–phonon mass enhancement factor. A similar work is undertaken by Ciuchi et al. [38]. Next we discuss high energy photoemission from extended levels. In principle the multiple scattering formula (51) can also be

173

applied to high energy XPS spectra. In contrast to UPS region, lmax amounts to about 30 in these high energy regions. For considerably large clusters, the multiple scattering calculations are impractical because the matrix inversion of the large matrices (1 − X)−1 is too time consuming. There electron scatterings from surrounding atoms are weak enough except the forward scatterings so that only low order scatterings are required for practical applications. The simplest approximation is to keep only the ﬁrst term of the multiple scattering series; Only the ﬁrst term in Eq. (45) is taken into account. The photoemission intensity Ik is written as the sum of the two terms, one- and two-center terms Ik1 and Ik2 Ik = Ik1 + Ik2 , Ik1 ∝

∗ Re c˛ c˛

ˆ L (k)M ˆ ∗ ML L , YL∗ (k)Y LL˛ ˛

LL

˛˛

Ik2 ∝ 2

∗ Re[c˛ cˇ exp[ik · (R˛ − Rˇ )] ×

ˆ L (k)M ˆ ∗ ML L ], YL∗ (k)Y LL˛ ˇ

LL

˛ˇ

(55) where atomic orbitals ˛ and ˛ are on the same atomic site ˛ whereas ˛ and ˇ are on the different atomic sites. The onecenter term is not inﬂuenced by atomic vibration because it has no information on atomic position. In contrast the two-center term has explicit information on the atomic positions in the exponential function. The position vector R˛ is written as the sum R˛ = R0˛ + u˛ , where R0˛ and u˛ are the equilibrium position and the displacement of the ˛th atom. Within the harmonic approximation, we can apply the Mermin’s theorem [39] because u˛ is linear in phonon operators. Applying this theorem to the thermal average over phonon states, we obtain an explicit formula for the two center term

Ik2 ∝ 2

∗ Re[c˛ cˇ exp[ik · (R0˛ −R0ˇ )] × exp

−k2 2˛ˇ

2

ˆ L (k)M ˆ ∗ ML L ], YL∗ (k)Y LL˛ ˇ

LL

˛ˇ 2

ˆ · (u˛ − uˇ )] . 2˛ˇ = [k

(56)

In the high energy region the thermal damping exp[−k2 2˛ˇ /2] is quite small, and the approximation Ik ≈ Ik1 ∝

|c˛ |2 ˛ ,

factor

(57)

˛

1 ˛ = √ |MLL˛ |2 , 4

(58)

L

works so well, where ˛ is proportional to the photoionization cross section of ˛th atomic orbital ˛ . This formula is called Gelius formula [40], and it is widely used for analyses of XPS valence band spectra in particular of organic molecules. In contrast to ARPES spectra in UPS region, we cannot obtain information about band dispersion because of strong thermal damping which destroys the interference between two photoelectron waves emanating from different atomic sites. The one center term is not inﬂuenced by the Debye–Waller factor, on the other hand the recoil effects have important contribution from the one center term in the high energy XPS spectra. At ﬁrst the recoil effects have been studied for deep core processes [41–43] and later for XPS from extended levels on the basis of the Gelius formula (57) [43–46]. We observe energy dependent binding energy shift and peak broadening which are successfully explained by phonon excitation caused by sudden recoiled motion of the X-ray absorbing atom.

174

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176

3.3. Extrinsic loss effects So far we have taken only the lowest order term in regard to the screened Coulomb interaction W into account, the ﬁrst term of Eq. (37). We next take the other three terms in Eq. (37), which are all in the order of W. Now we investigate the second term of Eq. (37) in more detail. The lesser part of GAıG contributing to the photoemission current has loss term, since it includes W< and W> . As shown by Eq. (25), both of their spectral representation have the delta function ı(ω ± ωn ), which contributes to the real losses. We pick up the loss term from the second term of Eq. (37) Ik ∝ Imfk− | g c (εk − ω)ı(1)< |fk− = Re

[fk− | g c (εk

− ω)vm |gn ×

fk− | m g c (εk

v

∗

+ ωm ) |gn × ı(εk + ωm − εn − ω).

To discuss the plasmon loss side band, considerably simple and reliable formulas are proposed for the ﬂuctuation potentials. By use of these potentials several authors have discussed energy and angular dependence of plasmon loss bands [1,2,48], where elastic scatterings of photoelectrons are neglected. From the detailed numerical calculations, although the interference should drop out as a function of photoelectron energy, the rate is very slow: even at 10 keV we still ﬁnd ﬁnite contribution from the interference [49]. In the low energy region the loss shape is quite different from that observed at high energy excitation. Neglecting elastic scatterings of photoelectrons, we can rewrite the simple formula for the sum Ik0 +

Ikm + double losses + . . .

m

nm

(59) In Eq. (42) we have εk = ω + εn (no loss), but the above formula provides the relation εk = ω + εn − ωm . This relation really describes the loss process with loss energy ωm . The propagator gc (εk − ω) at core binding energy is approximated by [47,4] g c (x, x : εk − ω) ≈ −

1

c (x) c∗ (x ). ωm

(60)

in an exponential form which is called cumulant expansion or Landau formula [13]. In high energy region, we can derive quantum Landau formula which describes the overall features of the photoemission bands: To recover the lowest order sum, the ﬁrst two terms in the above sum, and also satisfy the normalization condition, the formula (68) can be obtained [2]. Furthermore electron elastic scatterings inside solids can be discussed in purely quantum mechanics, so that important quantum effects such as interference can be studied [49,50].

This approximation simpliﬁes the amplitude as

fk− | g c (εk − ω)vm |gn ≈ fk− | |gn Sm , c |vm | c . ωm

Sm = −

(61)

We can neglect the contribution from gn (n = / 0) in Eq. (59) since |gn gn | has a factor |Sn |2 which should be small other than |S0 |2 . We thus obtain a simpliﬁed formula from Eq. (59)

Ik ∝ Re

ex [fk− | | c Sm m (k)∗ S0∗ ] × ı(εk + ωm − εn − ω),

(62)

m ex (k) = fk− |vm g c (εk + ωm ) | c . m

ex |m (k)S0 |2 ı(εk + ωm − εn − ω).

(63)

(64)

We should note that Eq. (42) can also describe the same ﬁnal hole state m* (65)

This is well known intrinsic loss (shake-up) photoemission intensity. Summing up all these four contributions, we obtain a formula to discuss the photoemission intensity for the loss band aside from the main peak. ex (k)S + f − | | S |2 × ı(ε + ω − ε − ω). Ikm ∝ |m c m m n 0 k k

(66)

This formula clearly shows the importance of the interference between the extrinsic (the ﬁrst term in | . . . |) and the intrinsic loss (the second term). On the other hand we can use a simple formula for the main photoemission band Ik0 ∝ |fk− | | c S0 |2 ı(εk − ω − ε0 ).

0

(68)

The function ˇ(ε) fully includes intrinsic and extrinsic losses deﬁned by ˇ(ε) =

|m (k)|2 ı(ε − ωm ),

ex m (k) = m (k) +

Sm . S0

(69)

(67)

(70)

It is often reasonable to split ˇ(ω) in a “low-energy” part ˇ1 (ω) and a “high-energy” part ˇ2 (ω). The low-energy spectral function D1 (ω) associated with ˇ1 describes X-ray singularity of the main photoemission band. The high-energy spectral function D2 (ω) associated with ˇ2 describes the plasmon losses taking both intrinsic and extrinsic losses into account, which can be expanded as D2 (ω) = ı(ω) + ˇ2 (ω) +

m

Ikint ∝ |fk− | | c Sm |2 ı(εk + ωm − εn − ω).

−∞

∞

dεˇ(ε)(e−iεt − 1) .

dt exp[i(ω + ε0 − εk )t] × exp

m

This term describes the interference between the intrinsic and the extrinsic losses. The propagator gc (εk + ωm ) describes the photoelectron propagation before the extrinsic inelastic scattering due to the ﬂuctuation potential vm : We thus understand that the ampliex (k) is the amplitude for the extrinsic loss 0* → m* . The tude m third term of Eq. (37) also gives the same interference term, and the fourth gives the extrinsic loss intensity: The explicit formula is given by Ikext ∝

∞

Ik∞ ∝ |fk− | | c |2

1 (ˇ2 ∗ ˇ2 )(ω) + . . . 2

where (A * B)(ω) = dω A(ω − ω )B(ω ) is the convoluten of A and B, As pointed out by Hedin et al. [2], only the intrinsic effects play a dominant role for the low-energy part (see Eq. (61)). ex (k), multiple scatTo calculate the extrinsic loss amplitude m tering approach is applied. Systematic calculations have been done for some simple metals [50,51]. Fig. 1(a) and (b) shows the depth proﬁles of single plasmon loss intensity associated with Li 1s photoemission for soft X-ray photons (ω = 170 eV): They show the proﬁle with and without elastic scatterings. The “extrinsic” uses only the ex (k) in Eq. (70), and “intrinsic” use only the second amplitude m term, “interference” uses their cross product for the calculations. The “total” shows the loss intensity by use of Eq. (70). The “intrinsic” shows exponential decay, whereas others show slower decay without elastic scatterings. All show smooth decays. On the other hand elastic multiple scatterings give rise to some prominent structure depending on surface structures, even though Li metals have only light atoms Li.

T. Fujikawa et al. / Journal of Electron Spectroscopy and Related Phenomena 204 (2015) 168–176

175

Fig. 1. The depth proﬁles of single plasmon loss intensity associated with Li 1s photoemission for soft X-ray photons (ω = 170 eV): (a) and (b) the proﬁle with and without elastic scatterings [51].

3.4. Resonant photoemission The real losses (extrinsic losses) are described in the formalism by use of the screened Coulomb interaction W> and W< . On ˜ describe the virtual losses as studied the other hand Wc and W before to analyze resonant photoemission including multi-atom resonant photoemission (MARPE) [21]. To understand the resonant photoemission on the basis of Keldysh Green’s function theory, we should use quantum electrodynamics (QED) [7], which clearly explains the physical source of the radiation ﬁeld screening [10]. If we build a photoemission theory on the semiclassical theory for the electron–photon interaction and within Keldysh theory, it is difﬁcult to explain the radiation ﬁeld screening ε−1 [10]. If we use different theoretical approaches like many-body scattering theory, we have no need to worry about this problem [13]. The radiation ﬁeld screening can also dramatically change UPS bands [10]. 3.5. Ultrafast photoemission X-ray free electron laser has opened a new research ﬁeld of photoemission spectra. A pump-probe photoemission technique is now a useful tool to investigate nonequilibrium dynamics of excited molecules and solids on a femto-second time scale. So far some interesting theoretical methods have been proposed to study the nonequilibrium dynamics observed in ultrafast photoemission spectra excited by pump laser pulse [52,53]. They have provided us with quite interesting information on electron dynamics in strongly correlated systems. These approaches are based on the intrinsic approximation where the dynamical coupling between the photoelectrons and the target are neglected: The photoelectron current is calculated by use of only the ﬁrst term of Eq. (1), so that extrinsic losses and resonant processes cannot be discussed in their treatment. Kazama et al. [54,55] have studied the sensitivity of the XPD angular patterns to the structural changes at rather high energy region (εp > 100 eV). They show that the XPD patterns are sensitive to the structural changes and the time-dependent XPD can be a promising tool to study nuclear dynamics after the laser pump excitation. We still have a question whether we can directly observe the time-dependent XPD spectra from the pump-probe XPS. Kuleff and Cederbaum have reviewed their recent theoretical results and some important applications after molecules are exposed to ultrashort laser pulses [56]. They focus on electron and nuclear dynamics after the excitation, in particular on charge migration. Detailed discussion on pump-probe XPS spectra has not been given there. In the pump-probe photoemission, we use two photon pulses. The pump pulse is so intense that it needs to be treated nonperturbatively, but the probe pulse is weak enough that A(1) (see Eq.

(33)) can be treated by perturbation theory. Within the intrinsic approximation, Freericks et al. have shown that the photoemission intensity can be given in terms of g< similar to Eqs. (40) and (41),

Ik (t) ∝ Im

d1d2fk−∗ (1)A(1)g < (1, 2)A(2)fk− (2)

(71)

where the upper limit of the time integration is t [52]. We should note that the lesser Green’s function g< has also information on the intense pump pulse. In the above formula, the time integrations of t1 and t2 are taken independently. A recent work on ultrafast photoemission theory using timedependent Dyson orbital for molecular systems is also quite promissing [57]. 4. Concluding remarks In this review uniﬁed theoretical analyses based on Keldysh Green’s function are demonstrated. In particular many-body effects due to electron–electron and electron–phonon interactions are studied in more details since they play important roles in organic solids: They are anisotropic and soft. In the UPS region ARPES analyses work because the Debye–Waller factor has small contribution to destroy the interference. In the high energy region Gelius formula and XPD analyses work well because of the strong cancellation of the interference terms. Do soft X-ray photoemission spectra give band dispersion? For hard material W, we can apply ARPES analyses even for X-ray ω = 870 eV [58]. Organic solids are soft, so that quite low photon energy can be a border for the turnover from ARPES to XPD. Acknowledgements This paper is dedicated to Professor Nobuo Ueno for his mandatory retirement at Chiba University, and for his great contribution to progress in organic solid science. One of the authors (KN) is grateful to the ﬁnancial support by the Career-Support Program for Woman Scientist at Chiba University. TF is grateful to the ﬁnancial support from a Grant-in-Aid for Scientiﬁc Research from Ministry of Education, Science and Culture of Japan, Project No. 25246041. References [1] W. Bardyszewski, L. Hedin, Phys. Scr. 32 (1985) 439. [2] L. Hedin, J. Michiels, J. Inglesﬁeld, Phys. Rev. B 58 (1998) 15565. [3] C. Caroli, D. Leder-Rozenblatt, B. Roulet, D. Saint-James, Phys. Rev. B 8 (1973) 4552. [4] T. Fujikawa, H. Arai, J. Electron Spectrosc. Relat. Phenom. 123 (2002) 19. [5] T. Fujikawa, H. Arai, J. Electron Spectrosc. Relat. Phenom. 174 (2009) 85. [6] T. Fujikawa, H. Arai, Chem. Phys. Lett. 368 (2003) 147. [7] T. Fujikawa, H. Arai, J. Electron Spectrosc. Relat. Phenom. 149 (2005) 61. [8] T. Fujikawa, J. Electron Spectrosc. Relat. Phenom. 136 (2004) 85.

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