The photon absorption edge in superconductors and gapped 1D systems
Opening of a gap in the low-energy excitations spectrum affects the power-law singularity in the photon absorption spectrum . In the normal state, the singularity, , is characterized by an interaction-dependent exponent . On the contrary, in the supeconducting state the divergence, , is interaction-independent, while threshold is shifted, ; the “normal-metal” form of resumes at . If the core hole is magnetic, it creates in-gap states; these states transform drastically the absorption edge. In addition, processes of scattering off the magnetic core hole involving spin-flip give rise to inelastic absorption with one or several real excited pairs in the final state, yielding a structure of peaks in at multiples of above the threshold frequency. The above conclusions apply to a broad class of systems, e.g., Mott insulators, where a gap opens at the Fermi level due to the interactions.
It was demonstrated more than years ago Mahan67 (); Nozieres (); review () that electron -ray absorption coefficient in metal, , is strongly modified by attraction to the localized hole left behind. The threshold behavior of absorption coefficient was found to be
In Eq. (1) and thereafter, stands for the difference between the photon energy and the core-hole energy measured from the Fermi level, and is the bandwidth. Prefactor, , contains the square of the dipole matrix element between the level and the conduction band. In the simplest case of a weak short-range attraction, , of electron to the hole the expression for the exponent has a form
where is the density of states at the Fermi level (we neglect the correction, , originating from the Anderson orthogonality catastrophe, Nozieres ()). Since the diverging absorption Eq. (1) comes from all energy scales between and , it is quite robust. In a finite system, the threshold behavior depends on additional energy scale, the level spacing Baranger ().
Interest to the singular behavior of near the threshold got a boost after it was predicted Matveev92 () that this behavior manifests itself in the resonant-tunneling current-voltage characteristics. This prediction was later confirmed in numerous experiments T1 (); T2 (); T3 (); T4 (); T5 (); T6 (); T7 (). Enhancement of absorption Eq. (1) was derived under the assumption that the density of states, , is constant within the entire frequency interval, . If there is a gap, , at the Fermi level the threshold behavior of is singular even without interaction with a hole:
and diverges near the edge of the gap. For small it could be expected Ma85 () that this strong bare singularity is weakly affected by the excitonic effects Mahan67 (). Indeed, the low-energy, , many-body processes across the gap, responsible for Mahan singularity, are suppressed. This reasoning suggests the form of the absorption in superconductor
Even stronger modification of the absorption spectrum takes place, when the core hole possesses a spin, so that the interaction with excited electron includes exchange. In this case two new physical mechanisms come into play. Firstly, a core hole creates in-gap states Rusinov () with binding energy measured from the edges. These states, in turn, affect dramatically the elastic scattering of excited electron transforming the near-gap absorption into
see Fig. 2. The absorption is zero at the threshold and resumes falloff only for . As a ”compensation” of the suppressed absorption, a -peak
emerges at the position of the bound state.
There is another many-body feature in , which is specific for the exchange interaction with core hole. This feature originates from the fact that exchange interaction of electron with localized magnetic impurity in metal can be accompanied by creation of an electron-hole pair KamGlaz (). The underlying reason is that localized spin emerges as a result of the on-site Hubbard repulsion of two electrons. On the other hand, with electron-electron interaction, two electrons can be excited by a single photon MishchGlazReiz (); DaiRaikhShah (). In the presence of a rigid superconducting gap, this process starts from the threshold MkhitRaikh () , which corresponds to inelastic absorption with electron and additional pair in the final state. This process is schematically illustrated in Fig. 1b. More additional pairs in the final state give rise to anomalies at , which have the form
Ii Derivation of Eq. (4)
ii.1 Time dependent superconducting Green functions
An efficient way Nozieres () to derive Eq. (1) is to consider scattering of excited electron by a transient potential, , and perform calculation in the time representation. In this representation the Green function of the normal metal ( or depending on ) has the form , where is the bandwidth. Generalization of the scattering approach to superconductor requires the time representation of the superconducting single-particle Green function
where is the spectrum of superconductor; with being the momentum measured from the Fermi momentum, , and is the Fermi velocity. The projection operators are matrices
with following properties: and . In the basis of eigenfunctions of the Bogoliubov-de Gennes Hamiltonian, interaction with the short-range potential is described by the diagonal matrix
Note that time-dependent Green function of a superconductor, obtained as a result of integration of Eq. (8), and subsequent summation over momentum, , can be conveniently expressed in terms of zeroth and first-order Bessel functions, namely
where the normal and anomalous Green functions, and , are given by
In the limit the normal-metal Green function, , is recovered from Eq. (12) by using the small- asymptote , while .
ii.2 Shape of the absorption edge
In superconductor, we generalize the response function to a matrix, , so that the absorption coefficient is given by the diagonal matrix element
As a result of matrix generalization, the expansion of the response function in powers of ,
In the normal metal, evaluation of is based on exact analytical result Nozieres () for the infinite sum
To arrive to Eq. (1) one has to set and , after which the square bracket in Eq. (II.2) reduces to , and integrate . Characteristic times in the relation Eq. (II.2) are arranged unevenly as illustrated in Fig. 3. The central interval is , so that are located in the close proximity, or (see Fig. 3) either to or to . It is important that in superconducting case the arrangement remains the same, and moreover, as we will see, and are always . This means that can be replaced by times the unit matrix. As a result, the matrix structure of drops out. The only Green function that retains the matrix structure is , Fig. 3. However, in the component , the anomalous Green function drops out, so that
where is defined by Eq. (12). Eq. (4) immediately follows from Eqs. (18) and Eq. (14). The Green function in Eq. (18) generates the density of states, , in Eq. (4). One point should be clarified with regard to the validity of the above result Eq. (18). We used the normal-metal solution Eq. (II.2). This is justified since integrals over , in Eq. (18) come from . This also validates the assumption , which we used to disregard the matrix structure of .
ii.3 Unconventional arrangements of times
There still remains a question whether or not the matrix structure of the superconducting Green functions, which becomes important near the threshold , gives rise to the contributions to , caused by ”unconventional” arrangements of times, , (), as shown in Fig. 4a and Fig. 4b; these arrangements are not relevant in the normal-metal case. For example, the simplest such ”unconventional” arrangement, Fig. 4a, manifests itself as an extra combination
where the is the asymptote of Eq. (12)
Note however, that the matrix structure in the integrand of Eq. (19) is
To integrate over , we perform multiplication of the first three matrices and obtain
Then the integration over in Eq. (23) reduces to
where we introduced a variable . Typical distance between the points, and , is , which suggests that the limits of integration can be extended to . Upon this extension we get
which is identical zero. The same reasoning rules out footnote1 () the more complex ”unconventional” arrangements of times at the threshold, like the ones shown in Fig. 4c. These arrangements, however, become essential in the case of exchange interaction with core hole, to which we now turn.
Iii Exchange interaction with core hole
Exchange interaction with core hole corresponds to replacement
where is a localized spin, and is electron spin operator. To illustrate the dramatic impact which the exchange interaction has on the near-threshold absorption, we return to Fig. 4a and corresponding expression Eq. (19). For potential interaction with core hole, this expression was identical zero by virtue of relation Eq. (22). Recall now that in the stationary problem the diagonal part of the exchange interaction, , creates two in-gap bound states Rusinov (): one below the upper edge by
and one above the lower edge by . The reason behind this effect is that effectively transforms the operator in Eq. (10) into the unity matrix. An immediate consequence of this transformation for our calculation is that the contribution Eq. (19) becomes finite. Subsequently, the contribution Fig. 4b and all higher-order ”unconventional” contributions illustrated in Fig. 5a are also finite. Within our formalism, the in-gap bound states emerge as poles, , of the Green function upon summation footnote () of infinite series of diagrams.
In deriving Eq. (5) for near the threshold, we in fact repeat all the steps which would render the stationary in-gap states. Namely, we notice that the phase of the integrand in Eq. (16) is large, which insures that the dominant contribution to comes from the domain , see Fig. 5a, when the net phase is ; contributions from the domains where are not ordered are suppressed by oscillations of the integrand. Thus we conclude that the integral Eq. (16) is dominated by . For the asymptote Eq. (21) to be applicable in this domain, the condition must be met. With ordered, the -fold integration in Eq. (16) can be carried out with the help of the identity
Depending on the parity of , the remaining integration, upon introducing the variables , reduces to
for odd , or to
for even . Finally we get
The product, , has a sharp maximum at , so that is large, which justifies the above assumption .
This expansion coincides term by term with the sum,
with given by Eq. (32). The sum over odd terms results in a simple exponent,
This exponent gives rise to the -peak, Eq. (6), in the absorption spectrum.
Iv Inelastic absorption
Up to now we neglected the spin-flip part,
of the exchange interaction. As it was mentioned in the Introduction, this spin-flip part of interaction between electron and core hole creates an effective electron-electron scattering KamGlaz (). This explains the possibility of inelastic processes with three quasiparticles in the final state, as illustrated in Fig. 1b. The threshold of inelastic process is . Here we will restrict ourself only to the behavior of inelastic absorption away from the threshold, , and follow the calculation in Ref. MkhitRaikh (). A great simplification away from threshold is that a ”golden-rule”- based calculation is sufficient. The rate of the process depicted in Fig. 1b is given by the following sum over the quasiparticle states with energies, , , and ,
where the first factor is the square of the amplitude, which is non-zero since the process involves a spin-flip KamGlaz (), and the dimensionless spin-flip coupling constant is
Near the threshold, , we have , , and . The matrix element near the threshold is approximately constant. This simplifies the summation in Eq. (37) to
Note that in the close vicinity of the threshold, , in-gap states created by the spinful core hole participate in the absorption, as illustrated in Fig. 1b. Namely, a pair of quasiparticles in the final state can consist, e.g., of one quasiparticle excited above the gap and empty lower in-gap state.
Our results Eqs. (4), (5) establish the threshold behavior of for a general situation when the density of states is strongly modified near the Fermi level but assumes a constant value away from the Fermi level. A notable example is a 1D interacting system. The shape of the Fermi-edge singularity in 1D interacting electron gas in the Luttinger-liquid regime has been studied in GogolinKaneGlazman () using the bosonization technique. Backscattering plays an important role in the exponent of the absorption. When backscattering opens a gap, the physics described in the present paper comes into play. The case of 1D Mott insulator near half filling makes the behavior of even richer, since the doping shifts the threshold. A related example is the Peierls insulator, when the charge density wave and ensuing gap at the Fermi level are due to electron-phonon interactions. Note, that in the latter case the gap is orders of magnitude larger than in superconductor.
Speaking about conventional setting for Fermi-edge absorption in metals, singularity in is smeared due to the finite lifetime, , of the core hole. In our consideration we assumed that the gap, , exceeds . In most experiments in metals the smearing of the edge is a fraction of eV, i.e., much bigger than a typical -value. However, the origin of this smearing is not a natural core hole lifetime broadening but rather a finite instrumental resolution LT79 (). The fact that observed absorption shape is a convolution of the singular , a Gaussian, which is measurement-related, and a Lorentzian, describing natural core hole lifetime, allows to separate the two contributions to the edge smearing. Early attempts LT77 () of such separation yielded meV for core hole. In the other experiment LT78 () involving core hole four times shallower than in Ref. LT77, , the natural width was found to be four times smaller, meV. In later experiment LT90 (), where the full broadening, meV, was very small, analysis of the data for the same absorption line as in Ref. LT78, revealed even smaller value of the core hole width in simple metals, meV.
As a final remark, the relevance of the exchange interaction of electron with the core hole was first pointed out in Ref. Girvin, .
We acknowledge the hospitality of the KITP at UCSB, where research was supported in part by the National Science Foundation under Grant No. PHY05-51164. We also acknowledge the support of DOE grant No. DE-FG02-06ER46313 (EM), Petroleum Research Fund grant No. 43966-AC10 (VM and MR), and NSF grant DMR-0906498 (LG).
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