Coupling of conduction electrons to two-level systems formed by hydrogen: A scattering approach
An effective Hamiltonian which could model the interaction between a tunneling proton and the conduction electrons of a metal is investigated. A remarkably simple correlation between the motion of the -atom and an angular-momentum change of scattering electron is deduced, at the first-order Born level, by using a momentum-space representation with plane waves for initial and final states. It is shown that the angular average of the scattering amplitude-change at the Fermi surface depends solely on the difference of the first two phase shifts, for small-distance displacements of the heavy particle. For such a limit of displacement, and within a distorted-wave Born approximation for initial and final states, the change in the scattering amplitude is expressed via trigonometric functions of scattering phase shifts at the Fermi energy. The numerical value of this change is analyzed in the framework of a self-consistent screening description for impurity-embedding in a paramagnetic electron gas. In order to discuss the so-called antiabatic limit on the same footing, a comparison with matrix elements obtained by the potential-gradient of an unscreened Coulomb field is given as well. The coupling of the tunneling proton to a free-electron-like electron gas is in the typical range obtained, by ultrasound experiments for different metallic glasses, from scattering rates for a Korringa-type relaxation process. That coupling is too weak to be in the range required for realization of the two-channel Kondo effect.
In the last decades hydrogens in metals deserved very extensive experimental studies and vast theoretical considerations. In crystalline solid the hydrogen () sits in a well-defined interatomic position like, e.g., in or . That is not the case in amorphous systems and at dislocations and other nonperiodic distortions. If the has some more room between the host atoms its position may be not well-defined and it moves between two positions. Such systems are known as two-level systems () and they have been very extensively studied (1). The coupling to the conduction electrons can result in an extra contribution to the electrical resistivity (2). If the atom has two metastable positions the electron scattering amplitude in different angular momentum channels depends on the atomic position. The difference between these amplitudes is described in the literature by a coupling . That coupling contributes to the resistivity in a conventional way.
There are, however, other couplings where electrons induce transitions between the two levels. Thus an assisted transition can be realized by the tunneling of the atom between the two positions (3); (4); (5). The importance of that coupling, denoted by and , is highly debated (6); (7). These models consider different atoms with sizeable differences in their masses. The original suggestion is limited to small tunneling rate, while more intensive tunneling induces an essential split between the energies of the atomic eigenstates which reduces their roles. The model has attracted considerable interest as it was suggested that at low temperature exhibits non Fermi liquid behavior known as the two-channel Kondo () effect, which has its own theoretical interest (8); (9).
Experimental supports come much more from the studies of point contacts than from direct measurement of the electrical resistivity (8). The Cornell group made detailed suggestion how the observed zero-bias anomalies could be due to that effect which has been also highly debated (6). Since that time the original model was modified (7) by taking into account the actual electronic structure at the . The possibility was also considered (10) where the atom moves between the two positions via the next higher energy level of the atomic motion. These suggestions were aimed to increase the electron assisted amplitudes (, ) to make the more feasible. The enhancement of these couplings by the renormalization due to the conduction electrons is drived by . The estimated couplings were on the borderline, therefore, those should be studied in more details. It is crucial whether those coupling strength could reach some certain regions which are very sensitive on the strength of .
Recently it was suggested that is a possible candidate (8). Indeed, zero-bias anomalies have been observed in hydrogenated point contact with considerable size (11). That system could deserve more extensive experimental studies. Such zero-bias anomalies were also observed in hydrogenated point contact (12). The idea has beeen also raised that other systems may contain some water with metastable positions of the hydrogen atoms.
These recent developments justifies further studies of the coupling of a heavy tunneling particle to conduction electrons. The satisfactory realiable estimation of these coupling could be performed by considering certain atomic configuration with detailed knowledge of the electronic density of states in the region of the . In other words, a material-specific description of the host could be based upon more realistic but much more complex specifications. Thus, a calculation of the coupling would be too ambitious in case where the structural surrounding would require more knowledge.
The present paper is devoted to a minimal program. An embedded proton is considered in a double-well potential which determines the two possible positions of the heavy particle. These positions are considered as stable ones. The main task is to take into account the screening, due to the Coulomb interaction and charge-response, in a satisfactory way. In this work the ground-state screening in the surrounding electron gas is described in a self-consistent Hartree-like manner, by using Kohn-Sham independent single-particle states whose occupation is prescribed by the Pauli principle. The change in the scattering amplitude, due to a potential-displacement, is expressed in our renormalized perturbation theory via matrix-elements taken between stationary scattering eigenstates of the external Kohn-Sham field, instead of commonly applied matrix-elements between unperturbed plane-wave states of a system without impurity. Concretely, in the present work we shall apply the distorted wave Born approximation (13) for scattering characteristics.
The paper is organized as follows. The next section, Sec. II, is devoted to the details of our theory and a discussion of the plane-wave-based, first-order Born approximation for matrix-elements is also given. These results, obtained by using screened potentials, are compared with those based on an unscreened Coulomb field. The last section, Sec. III, contains a short summary and an outlook for further possible developments. The Appendix summarizes the commonly applied basic elements of previous motivating works.
Ii Theory and results
As we motivated in the Introduction, an atom is considered which has two stable positions in metallic matrix due to a double potential well. The positions are symmetric to a central point and they are at , respectively. This is the standard picture to a theoretically active field with considerable experimental relevances; see Figure 1 for illustration.
The main goal of the present contribution is to determine the change in the electron scattering amplitude in different angular-momentum channels when the atom is moved out from the central point to one of the two positions, but the set of the electron wave functions is still centered at the origin. These positions, at as the illustrative Figure shows, are described by a pseudospin .
The atom is embedded in the degenerate electron gas of the metallic target. The general form of the Hamiltonian is , where the diagonal matrix stands for independent electrons in stationary eigenstates of a self-consistent external field
in which the are energy eigenvalues of bound and scattering eigenstates. The and create and annihilate these eigenstates of Eq.(1) of spin .
In order to show the difference of our distorted-wave Born method from the commonly applied plane-wave approximation, and thus provide a clear phenomenology to understanding, we start by assuming a weak potential-energy and write in this unperturbed-state representation for initial and final states as
with the corresponding operators for creation and annihilation of plane-wave states. Here we have , and thus is given by
Since the perturbation , due to the shift in the atomic position, is real we have the character. In other words, the is Hermitian. The change in the scattering amplitude is . The extended continuous system can be recovered simply by using the prescription . The -summation must be performed at the Fermi level to respect the Pauli’s principle, and thus the determinant-character (14) of the state vector of . This will result in, by using the exact expression for the momentum change in elastic scattering, an averaging (see below) over the scattering angle. From this point of view, our procedure is similar to the one (15); (16) used in the theory of conventional superconductivity to define a repulsive (electron-electron) pseudopotential, from a spherical at the Fermi surface.
One can calculate the change  in the scattering amplitude beyond the usual Born approximation, by using expansions for properly defined (13) initial and final states in terms of normalized spherical harmonics. Furthermore, it was suggested (4) that one can define a Hermitian , beyond the weak-coupling limit, in the following way
where is a suitable (4) coupling, mediated now, in our case, by a perturbation. The intruduced (5) functions are spherical harmonics, and and run over a properly chosen set (see below) of angular momentum indices . The present calculation is carried out in several steps.
Calculations of the screened potential around the proton and the phase shifts. The charge is sitting in the central position where its screened field and a complete set of one-electron wave functions of occupied states are determined in a self-consistent way. This is achieved, in practice, by applying the Kohn-Sham method of density-functional theory (), which reduces the complicated many-body problem of the inhomogeneous electron gas (in the presence of a charge) to a single particle problem (17). The calculations are performed for a grand-canonical system, i.e., at a fixed chemical potential, applying local-density approximation () for the exchange-correlation potential. The single-particle potential energy has a simple form in this approximation
in which is the screening density. The many-body term is expressed via an input exchange-correlation chemical potential as , in order to have a vanishing effective potential energy at infinity.
For a given density () of the screening environment, and depending on the magnitude of the attractive embedded charge, the total density consists of bound and scattering eigenstates. The total density, the basic variable of , is constructed out by summing over doubly occupied bound and scattering-like states
in which are self-consistent solutions of the radial Kohn-Sham equations with at scattering energy, and where is the phase shift. The above continuum states are normalized on the -scale, and thus the scattering () part of the induced density comes from an integral over the Fermi-Dirac distribution function
The total-screening condition, , implies the Friedel sum rule of scattering phase shifts in one-electron mean-field treatments
The rule is satisfied, of course, at numerical self-consistency of iterations. Here denotes the number of occupied bound states. In a ground-state calculation, on which the present work is based, the last two terms cancel each other according to Levinson’s theorem (18). At metallic densities, already the first few phase-shifts provide a very accurate approximation. For example, at the value of the density parameter one has and in the case of a proton. This effective can characterize the mobile part of the electron fluid of a target, and satisfies (19) a necessary condition appropriate for correctly describing ground state properties of defects.
Illustrative results are exhibited in Figure 2 for the leading, and , radial wave functions calculated at the value of the Wigner-Seitz parameter. The component shows a Coulomb-like enhancement, at the origin , over the plane-wave-based unity. The deviations from the plane-wave components and are notable, as Figure 3 shows via the corresponding products of components. The product has its maximum at with value , and it becomes zero at =2.53. For higher values it oscillates with decreasing amplitudes around zero due to (shifted) Friedel oscillations. As Figure 3 shows, the perturbative product is much more extended. Its maximum (0.26) is at about =1.7, it becomes zero at about =3.9, and has Friedel oscillations beyond this. Clearly, the attractive proton is a strong perturbation at metallic densities.
As a digression we enumerate at this point few important facts which support the reliability of the above phase shifts and the radial functions even in a real lattice where the structural (atomic) surrounding could have influence. First, even unit charges () represent strong local perturbations. This is verified by experiments for slowing down of low-speed () protons and antiprotons moving through paramagnetic metallic targets characterized by , where . In the theoretical (20) energy loss per unit path length  appears the transport-cross section
The theoretical results, based on phase shifts for protons (21) and antiprotons (22), are in impressive agreement with -data obtained (23) at dedicated facilities of CERN and show, via the ratio (), a pronounced () charge-sign effect which rules out (24) the applicability of a simple Born-approximation. For completeness, for an antiproton one has =0.7729 and =0.2003 at . The illustrative Figures for the radial wave functions, and the above-outlined phase shifts values for embedded unit charges () suggest [see the discussion at Eq.(24) also] the use of the so-called -approximation in order to get a reasonable estimation for our coupling constant. This will be, therefore, the practical approximation in the problem.
Quite remarkably, a dimensionless parameter , which expresses (25); (26) the overlap of two many-body ground states with the same local potential at two sites separated by a short distance , depends on the phase shifts similarly
where is the de Broglie wavelength of an electron moving with velocity; . Related to positive muon () quantum diffusion in metallic targets, realistic phase shifts were already applied (27) to estimate this dimensionless parameter.
Second, while the band-structure paradigm emphasizes the importance of lattice structure, calculation (28) for the system using a molecular-cluster model shows that the electronic properties of the impurity are dictated mainly by its local environment. The such-calculated on-top screening density, , is only slightly smaller than the simple jellium-based result which tends (18) to from above as decreases. Furthermore, the molecular-cluster and pseudo-jellium calculations for are in close agreement with each other for -displacements up to from the equilibrium configuration. A self-consistent pseudopotential calculation (29) shows that the proton is screened on one atomic distance, and the hydrogen always has more charge around it than the upto the Wigner-Seitz radius. Note, that the amount of electron-localization in -screening could characterize, as was pointed out (30) recently, the site-preference of hydrogen in new storage materials.
We finish our supporting enumeration by associating the above local-environment picture with the proposal of Hopfield (31) for short-range properties when there is a change in potential due to moving an atom by a small distance which results in matrix-elements needed, in his case, to an estimation of electron-phonon coupling in transition-metal superconductivity. Namely, it was shown that when an angular momentum decomposition of electron wave functions is used, the matrix elements contain chiefly scatterings which change the angular momentum of the electron. Precisely, it is this character which is central in the context of scattering of electrons from two-level systems. The parity change of the angular momentum state (without altering the spin indices) of conduction electrons gives them an internal degree of freedom coupled to that of the impurity (4); (32). This internal degree is the background to establish an analogy with the usual (spin-related, magnetic) Kondo effect.
The change of the potential for . The screened charge is moved to one of the positions carrying the potential, which is taken rigid as the screening action is very fast (33) compared to the infrared processes essential in some other problems. The electron eigenfunctions are also carried with the tunneling atom, but they are decomposed is terms of those have already been determined for the central position; see in point . Thus the new potentials are
where the important (perturbative at small ) term is given by
A Taylor expansion in results in a simple dipolar form in the leading order
We illustrate the behaviours of the self-consistent potential and its gradient in Figure 4. Concretely, the and products are plotted by solid and dashed curves, respectively. The inset is devoted to the finer details of these important functions. The calculated results refer to for the density parameter. The remarkable Coulombic character of at short distance is due to a compensating effect between the electrostatic (Hartree-term) screening and the local exchange-correlation term.
In the following perturbation theories are applied in terms of . The matrix elements are calculated between the states with wave functions determined in the previous section () but with the plane wave functions are also presented. Namely, we use the distorted wave Born and the usual Born approximations, respectively.
Matrix elements of the shifted potential between the original wave functions. In order to get a convenient, dimensionless (5) coupling to characterize the effect of the perturbation on precalculated continuous states we shall use energy-normalization (34) for these:
The original matrix element at is multiplied, in such a way, by the density of states per unit volume for a given spin evaluated at the Fermi energy. Notice, oncemore, that the change in the scattering amplitude [i.e., the matrix element of the perturbing between initial and final states (13)] should involve a prefactor. With Eq.(13) the angle-integration over gives, by applying standard recurrence relation for , the following simple result
Only the and values are allowed, due to the dipolar character.
Born approximation. It is instructive first to study the case of a weak, pseudo-potential in Eq.(13). Thus, we perform the radial integration in first-order Born approximation, i.e., we apply plane-wave components for the free radial wave functions as
In this perturbative case the integration by parts and use of the following expression based on recurrence relations for Bessel functions
results in a remaining integral now with . In such a way we can apply the standard definition (13) of the first-order Born () phase-shift
to obtain the following informative [see also Eq.(26), below] expression
We stress that this equation is valid, physically, only for small values of the phase shifts, i.e., when the distortion of the electron wave function by the central potential field is negligable.
If a weak potential is given via its Fourier representation, as in a dielectric screening of the external field , the Born phase shifts (we use atomic units here) are
where is the standard hypergeometric function. In our simple, so-called -approximation, one can get from Eq.(20) easily [see Eq.(24), also] the leading difference-term
This -space representation could be useful if the inverse Fourier-Hankel transfomation results in a complicated function for , and a fast estimation is needed. For a commonly applied Yukawa-type potential, , the Born phase shifts from Eq.(18) are given by Legendre-functions  of the second kind as
It is important to note, in the present context, that a restricted Friedel sum
still holds despite the perturbative approximation if , i.e., the screening parameter corresponds to the quasiclassical Thomas-Fermi value. Physically, the normalization of the screening charge (calculated from the model Yukawa form via Poisson’s equation) is satisfied, but its real-space distribution is not necessarily realistic.
The first-order Born phase shift does not contain the multiple scattering effect in the central potential field. Formal applications of Eqs.(18)-(19) with strong potentials might result in an uncontrollable estimation for the effect in question. However, the proposed correlation (4) between the motion of the TLS-atom and an angular-momentum change is now transparent even at the first-order Born level, as Eq.(19) clearly shows. Note that this transparency of correlation is obtained (easily) when one applies (31) spherical-harmonics-based expansions for the initial and final (plane waves in first-order Born approximation) states separately, i.e., by implementing Eq.(4).
When, still in first-order Born approximation with a in Eq.(13), we implement Eqs.(2-3) with the standard spherical-harmonics expansion for , i.e., without the mentioned separation for initial and final states, one gets () for small enough
In this representation of the (perturbative) matrix-element a parity-change is not transparent. But a partial integration and the use of the recurrence relation gives for the integral. The Fermi-surface average, ( with , becomes, as Eq.(21) shows, a function of  solely. This conclusion agrees with Ref.; only the and harmonics are relevant.
Beyond the first-order Born approximation. In a quite recent theoretical work (7), which also rests on matrix element calculation with a potential-gradient between normalized and bound-states, the bare  Coulomb potential was applied to characterize in a metallic matrix. Here, by using the Sommerfeld parameter, we add the corresponding exact phase-shift difference for the case of a Coulomb field
By using the accurate and radial wave functions to integration in Eq.(16) above [and not and to Eq.(18)] with gradient of the true behind these functions one obtains, based on earlier result (35); (36), the exact
closed expression in terms of scattering phase shifts [see: Eqs.(9) and (10), also]. The exact result, which contains now the multiple scattering effect in the central potential field to all order, is a bounded function in contrast to Eq.(19). We illustrate, in Figure 5, the argumentum-function of Eq.(16) by using the self-consistent solutions at . Fortunately, as noted above, we have already a simple, analytic result in Eq.(26) for the integral without further numerics.
We add at this important point that with Eq.(25) and Eq.(26) one can get the
expression in terms of the parameters , , and , for a Coulomb field; . The plane-wave-based perturbative Coulomb limit, obtained with to Eq.(16) is surprisingly similar to Eq.(27). The difference is only, but this is crucial since in a.u., that there is not term [see Eq.(21) also] in the denominator. Formal use of a finite to Eq.(27) could mimic a screening-regularization and, together with Eq.(15), could thus allow fast analytical estimations. For the Coulomb case in the limit (formally ) one has , where is the electron velocity. Notice, parenthetically, that this difference is one of the pedagogical examples in physics which shows transparently the role of Planck’s constant, , ”in action”.
After the above detailed analysis on scattering, we return now to the matrix element needed to a suitable (4) coupling introduced in Eq.(4). First we give, by using the previous notations, a dimensionless expression related to the scattering-amplitude change
The obvious connection in the weak-perturbation limit is, based on the peculiarity of the first-order Born () approximation with plane waves, simply .
This transparent connection between physical quantities at the Fermi surface suggests us to use, beyond the above weak-coupling limit, the extension. With our choice for boundary conditions to select initial and final states (involved in matrix-element calculation based on the distorted wave Born method) this seems to be the only logical step which preserves the important Hermitian character of and reproduces the weak-coupling limit. In addition, the proposed extension is in harmony with standard textbook statement (37) on the characterization of an energy shift of a particle interacting with a potential. Considering the above-deduced, remarkably simple rule on the true effect of an angle-averaging at the Fermi surface [see, at Eq.(24)], we write
for the important real () part, and the imaginary () part which is at least second-order in a weak-perturbation () limit.
Our physically-motivated extension to fix a value to the coupling of a with conduction electrons, remains within the framework of common knowledge: All stationary properties of metals which can be described in terms of scattering of conduction electrons based on an adiabatic picture, are periodic functions of the phase shift, except, curiously, the Friedel sum rule. We stress that in our model an electron merely sees a different scattering potential for each state of the configuration but has no effective internal-spin degree of freedom. The states, for initial and final states, are needed (13) to a perturbation theory in the continuum of a given potential field. In experiments for transport characteristics, we have current-carrying electrons. Thus a standing-wave representation (which would refer to a different boundary condition described by physics, for example by a cavity) for the states involved in our matrix-element calculation is not reasonable.
In the so-called unitary limit, , where the effect of the self-consistent central field is strong [see, Figs. 2 and 3], the influence of the potential-shift [ becomes very small, i.e., is small. Remarkably, this statement is in accord with the conclusion of Györffy on a renormalized limit (obtained via mapping to the partition function of a fictitious, auxiliary logarithmic gas) with contact interaction (38). This shows that the ground-state charge-distribution around an embedded proton is, in fact, intact to small extra fields. In a one-phase-shift approximation the maximal coupling would be at the mathematical value. At this value the real and imaginary parts of the scattering amplitude are equal in magnitude. We shall return, briefly, to the simple contact-potential approximation in the Appendix.
The expression in Eq.(28), which gives a strict linearity in , is based on a leading-term expansion for the perturbation as Eq.(13) shows. In order to get a more detailed -dependence of coupling due to the displacement of a screened proton in the -direction, we performed numerical volume-integrations with Eq.(13) and the dominating product. By introducing the simplifying notation
motivated by the need (see below) of explicit estimation for a physical observable, and performing the -integration via a variable-change, we have
The numerical result for is presented in Figure 6 by a solid curve. The dashed curve refers to the asymptotic expansion, which is linear in .
We note that the asymptotic expansion provides a quite acceptable representation upto about , and even at the deviation from the numerical results is only about . The somewhat surprising linearity can be explained, partly, by the fact that the second term () of a formal Taylor series for the perturbation would give vanishing contribution, i.e., there is not quadratic, -proportional, term at small in the approximation. Beyond the physically reasonable value, the function grows more and more gradually and it has a maximum at about . With an acceptable mathematical accuracy one may use the fit for in our -dominated problem.
Finally, the magnitude of can be calculated by using our phase shifts at density parameter [in this case ], and assuming (see Fig. 6) a size of the TLS and then
That value is moderate enough to ignore the multiple scattering due to (see Appendix B), and is in accordance (2); (39) with the estimated typical values for different metallic glassy systems in the intermediate coupling regions using ultrasound measurements. In our modelling of the amplitude-change due to a , the absolute maximum at does not provide more than an about increase. A naive mixed approximation, in which the gradient of a bare Coulomb potential is weighted by our self-consistent product in Eq.(16), gives a smaller numerical value than Eq.(26). This is due to, mathematically, an over-weighting of the negative part of the product in Fig. 3 by an unscreened-field gradient in the volume-integral. A careful discussion of the underlying adiabatic and anti-adiabatic pictures was given by Kagan (40), by stressing the relevance of an adiabatic character due to the fast screening-action, in -motion.
Iii Summary and outlook
A dimensionless coupling constant that characterizes the effect of a potential-gradient perturbation on scattering eigenstates of a self-consistently treated embedded impurity is deduced for small values of the impurity displacement in metallic electron gases. The result is expressed via bounded, trigonometric functions of scattering phase shift differences at the Fermi energy, quite similarly to a well-known overlap parameter . Beyond the leading-term expansion for the perturbation, the numerical results show that the coupling parameter has an almost linear -dependence upto the physically realistic value for the displacement of a screened proton in an electron gas with .
As we mentioned earlier, at the enumeration of supporting facts related to the applied self-consistent-field method, it is the parity change of the angular momentum state of conduction electrons which gives them an internal degree of freedom in the problem. We derived this parity change in the present paper by using a distorted wave Born approximation, i.e., using those scattering wave functions to matrix-element calculation which are determined by the central field of a fixed, spherically screened proton. Beyond the applied distorted wave Born approximation, with an axially symmetric scattering potential, the exact description would lead to coupled radial equations (41) in a partial wave expansion of the scattered wave at a given . Particularly, the -dependence of the coupling beyond the present renormalized perturbation theory with precalculated spherical states, is an exciting problem. Application of internally consistent (axially symmetric) nonperturbative wave functions to the problem needs a future study. The numerics could be based on a -method developed recently (42) for axially symmetric potentials in an electron gas.
It is generally true that a clean theory, like the present one based on scattering aspects, is an analysis of the properties of an idealized, hypothetical model. What we have tried to do is to give a self-contained mathematical treatment of a physically motivated model, hydrogen in an electron gas, that can demonstrate the strong local effect in an atomic displacement in a metallic target. A more detailed, i.e., material-specific description of the host could be based upon more realistic but much more complex physical specifications. For example, simple application of a specified, as a function of the hydrogen concentration, density of states at the corresponding Fermi energy of a real system could change the numerical value of the dimensionless coupling, found here with a free-electron form .
Our self-consistent Kohn-Sham approximation incorporates the electron-electron interaction in the Hartree-like mean-field potential which acts for independent electrons. Beyond this approximation, and especially with bound states around the bare impurity, the treatment of electron-electron interaction (statistical and dynamical) has to be somewhat more sophisticated, since part of it is already included in the screening. One must avoid double counting, and consider self-interaction corrections. A strong interaction between the electrons, involved in the screening action, may well lead to the occurrence of localized magnetic moments associated with an embedded impurity atom. In such a cases the interplay between the orbital and spin degrees of freedom may become an important question.
As we fixed in the present paper, in our model an electron merely sees a different scattering potential for each state of the (impurity) configuration but has no effective internal-spin degree of freedom. A nontrivial extension of the applied method, i.e., the distorted wave Born approximation, could be a problem where in addition to our displacement there is an extra spin-orbit () coupling already at the level of our via a scattering interaction: , where . As was demonstrated (13) by Ballentine, the change in the scattering amplitude has a peculiar character in this case. Namely, there is no term with in the amplitude-change, because the operator yields zero in that case. How a combined perturbation could change the statement obtained at Eq.(24) in a problem might deserve a detailed study.
The theoretical modeling of important observables based on transport-related experiments needs additional care. For example, a recent calculation shows the possibly important role of different charge-states in current-driven electromigration and backflow, (43) where the long-range characters of scattered waves are more important than in our present problem for a coupling. We have discussed and emphasized the importance of physical boundary conditions (which appear in the fundamental Lippmann-Schwinger equation) for continuous states which are based on a second-order differential (Schrödinger) equation. If our displacing-atom were embedded into a system described by externally generated (44) standing waves, one could use (37) the normalization-change and, thus, get for the right-hand-side of Eq.(26) only the difference of . In this case, the coupling could enhance almost resonantly. Indeed, the mentioned study (44) heralds, via spectroscopic informations, a common nonmagnetic effect with different impurities in standing-wave patterns. We speculate that the dissolution of hydrogen into -electrodes can make a local confinement-like effect for states involved in the conductance (11) of nanojunctions.
Finally, a way to consider electronic inhomogeneities can be an additional local-density approximation governed by the strong short-range distortion on which our model is based. Between atoms of a real lattice there are ranges, measured from a lattice-atom position, where the density of states has enhanced local values. If the allowed tunneling occurs in such geometrical ranges, a mathematical averaging of our function over a certain range of seems to be reasonable. The role of an almost ferromagnetic nature of a real target (expressed via a Stoner-enhancement (45) in the spin response function) might also deserve future considerations. These combined considerations could give further quantitative information to a field of considerable experimental relevance.
Acknowledgments. The authors thank L. Borda, R. Diez Muiño, and G. Zaránd for useful discussions. A.Z. is grateful to the Humboldt Foundation to support his stay in Munich, where part of this work was done. I.N. acknowledges the warm hospitality at the DIPC, San Sebastián. The work has been supported partly by the OTKA: Grant Nos. T046868 and T049571 for I.N., and Grant Nos. T048782 and TS049881 for A.Z., respectively.
Iv Appendix A: Connection with earlier theories
This Section is devoted to a comparison with previous works (5); (46); (9) on modeling the coupling in Eq.(2), based on plane-wave states. In order to provide a clear phenomenology, we stress the point that the present theoretical description, which is based on calculation of matrix-elements of a dipolar potential-perturbation, also uses prefixed basis sets. Namely, plane-waves in the conventional and precalculated self-consistent ones in the distorted-wave Born approximation for continuous states.
In the main text, at the details of the first-order Born approximation, we outlined the -representation of the coupling. Thus the desired link, for example to Ref. with a Yukawa potential, is easily obtained by using the scattering value of and the
Born-representation to Eq.(24) for the quantity; . Motivated by certain scattering-length arguments, the notations were adopted in the mentioned earlier works instead of the ratios. Somewhat fortuitously, the numerical value of the perturbative difference in Eq.(19) with a Yukawa-type () screening
is not far, at least for , from the precise value obtained from Eq.(26) with self-consistently determined phase shifts of an embedding problem. Qualitatively, the mistake one makes in choosing a linearly screened potential is ”compensated” by the use of the first-order Born approximation; neither of these approximations are quantitatively accurate.
More importantly, the careful numerical analysis of Ref., performed by assuming higher () angular-momentum channels (i.e., using more parameters) to an estimation on -coupling, shows that (contrary to naive earlier expectations) there is not a series of Kondo-like effects corresponding to the increase of different orbital channels. This, numerics-based statement, is in harmony with the rule established at Eq. in the present paper.
A second-order Born approximation [valid for ] for the scattering (transition) amplitude  with the above simple Yukawa potential gives
in atomic units, for simplicity. This approximation with a fixed, linear-response-based input potential would suggest an enhancement of the coupling. A more consistent () treatment, in which the linear screening is also modified upto the second-order by a quadratic-response method, reduces (24) this enhancement as follows
showing that still holds for , but in the forward () limit . This observation, which is based on selected (RPA) diagrams, heralds that care is needed when one uses a higher-order method in terms of a bare (in our case: a linearly screened) input potential in field-theoretic many-body attempts.
A central local potential, for example , gives rise to scattering of all orders of spherical harmonics even at the first-order Born level. On the other hand, a so-called separable potential (47); (48) for a given orbital () momentum causes scattering only for the given (th) partial wave; in the case of all , one speaks of a completely separable potential. A correct determination of the corresponding channel-potentials, , in this modeling could rest on experimental data or on a detailed, microscopic theory.
It is important from the point of view of physical consistency, that the frequently applied tangent-method (49) on multiple scattering effects gives (48) a in terms of , , and of the -channel potential. In the even simpler case with a contact auxiliary interaction (which gives a constant potential in momentum space) one has the remarkably simple form for the -wave phase shift. Note that by writing, formally, in the lhs of Eq.(18), one can get this exact result in one-step from
The precise derivations of the exact result rest on much more involved calculations by using real-space (50) or momentum-space (51) Schrödinger equation with . Application of a renormalized contact interaction, defined via , in the perturbative rhs of Eq.(39) results in an identity. Clearly, this renormalized interaction could be used with unperturbed plane wave states to calculate the real reactance-matrix. The exact standing () wave solution [cf. Eq.(45)] of the tangent-method is the following
This is based, oncemore, on the principal-value Green’s function.
The standard logic to determine a value of is based on the phase shift  of the real potential , but the -channel contact interaction (10) has an important limitation when we apply it to the screening problem of a charge . It was pointed out earlier (52) that this model cannot supply enough charge to shield the Coulomb field of the physically simplest impurity . A formal requirement of would result in , when the self-consistent  leading phase shift (for ) goes to . At of the Wigner-Seitz density parameter, an about value (in a.u.) is prescribed by this formal constraint.
The auxiliary contact-potential was applied earlier within a local-phonon model of electron-phonon interactions (51) in an opposite way, i.e., via a direct approximation () for it to characterize the (constant) momentum-space potential of an atomic displacement in a free electron gas. In this case could measure a renormalized effect of a prefixed input , beyond the conventional first-order Born approximation (). Motivated by that work, the scaling equations of the two-level problem has been investigated (53); (54) in details in terms of an input variable in the so-called large-phase-shift case. The mentioned direct approximation was implemented (7) recently as with a dimensionless matrix element () of the Coulomb-potential gradient of an embedded proton taken between the corresponding hydrogenic ( and ) bound states, and weighted by an enhanced density of states due to an other model potential; cf. Ref..
The scattering with contact interaction is isotrop, i.e., the scattering amplitude (and thus the diagonal transition -matrix) does not depend on the scattering angle. The corresponding, dimensionless  exact scattering amplitude is
Its real part has the simple form of
which could suggest, based on a proper reinterpretation of the rhs, an effective (contact) potential to calculations with unperturbed plane wave states. If the impurity scattering is purely local one can use a many-body Green-function-based method (56) also to derive the exact result in Eq.(41), as the -matrix depends only on energy and not on momentum; such a derivation shows the algebraic nature of the propagator method.
For a single impurity embedded in an electron gas, and described by a regular potential, the electron Green’s function can be written symbolically as
using the standard -matrix approach (56). In our case the bare Green’s function of a free electron of the ideal system with chemical potential is . Since the Fourier transform of a contact interaction is , the equation for the corresponding -matrix is particularly simple as it depends only on energy and not on momentum
One easily gets the solution, where the quantity is the local propagator at the impurity site averaged in the momentum-vector and taken at the Fermi surface, and . Thus one has (56) the simple