arXiv subjects
Publications and source records attributed to Gerd Bergmann.
The calculation of the Kondo cloud is extended to disordered hosts. For a weakly disordered large three-dimensional host the structure of the ground state is very close to the pure host. However, the range of the disordered electron basis is much shorter. The extention of the Kondo cloud is essentially given by the square root of the Kondo length in the pure host times the mean free path.
The Kondo ground state has been investigated by numerical and exact methods, but the physics behind these results remains veiled. Nobel prize winner Wilson, who engineered the break through in his numerical renormalization group theory, commented in his review article "the author has no simple explanation ...for the crossover from weak to strong coupling". In this article a graphical interpretation is given for the extraordinary properties of the Kondo ground state. At the crossover all electron states in the low energy range of k_{B}T_{K} are synchronized. An internal orthogonality catastrophe is averted.
The multi-electron wave function of an interacting electron system depends on the size of the system, i.e. the number of electrons. Here the question investigated is how the wave function changes for a symmetric Friedel-Anderson impurity when the volume is doubled. It turns out that for sufficiently large volume (when the level spacing is smaller than the resonance width) the change in the wave function can be expressed in terms of a universal single-electron state |q> centered at the Fermi level. This electron state is independent of the number of electrons and independent of the parameters of the Friedel-Anderson impurity. It is even the same universal state for a Kondo impurity and a symmetric Friedel impurity independent of any parameter. The only requirement is that the impurity has a resonance exactly at the Fermi level and that the level spacing is smaller than the resonance width. This result clarifies recent fidelity calculations.
The Friedel oscillations in the vicinity of a Friedel-Anderson (FA) impurity are investigated numerically. For an FA impurity in the local moment limit the normalized amplitude A({\xi}) is S-shaped, approximately zero at short distances, approaching two at large distances and crossing the value one at the characteristic length {\xi}_{1/2}. Surprisingly, the Friedel oscillations of a simple non-interacting Friedel impurity with a narrow resonance at the Fermi level show a very similar behavior of their amplitude A({\xi}). A comparison correlates the resonance width and the Kondo energy of the FA impurity with the characteristic length {\xi}_{1/2} of the Friedel oscillations.
The fidelities of the Kondo and the Friedel-Anderson (FA) impurities are calculated numerically. The ground states of both systems are calculated with the FAIR (Friedel artificially inserted resonance) theory. The ground state in the interacting systems is compared with a nullstate in which the interaction is zero. The different multi-electron states are expressed in terms of Wilson states. The use of N Wilson states simulates the use of a large effective number N_{eff} of states. A plot of ln(F) versus N\proptoln(N_{eff}) reveals whether one has an Anderson orthogonality catastrophe at zero energy. The results are at first glance surprising. The ln(F)-ln(N_{eff}) plot for the Kondo impurity diverges for large N_{eff}. On the other hand, the corresponding plot for the symmetric FA impurity saturates for large N_{eff} when the level spacing at the Fermi level is of the order of the singlet-triplet excitation energy. The behavior of the fidelity allows one to determine the phase shift of the electron states in this regime. PACS: 75.20.Hr, 71.23.An, 71.27.+a, 05.30.-d
The definitions of the Kondo energy in the numerical renormalization group (NRG) and the Friedel artificially inserted resonance (FAIR) theory fail sadly for small samples where their predicted Kondo energy increases, while in reality the Kondo effect disappears. Therefore a different, universal definition of the Kondo energy is proposed, which uses the evasion of the orthogonality catastrophe by the Kondo impurity. In the absense of the Kondo effect the multi-electron scalar product (MESP) between all occupied spin-up and spin-down states approaches zero (the so-called orthogonality catastrophe). In contrast in the Kondo ground state the corresponding conduction electrons of opposite spin are pairwise aligned within the Kondo energy. In the present paper the MESP is investigated for the FAIR solution of the Friedel-Anderson impurity. The MESP is numerically determined for the (enforced) magnetic and the singlet states as a function of the number N of Wilson states. The magnetic states show an exponentially decreasing MESP as a function of N. In the singlet state the ground state requires a finite MESP to optimize its energy. As a consequence there is no orthogonality catastrophe. Within the energy range of the Kondo energy the scalar product between corresponding (single electron) spin-up and spin-down states is very close to 1.000 and falls off beyond the Kondo energy. The energy which separates the two regions is well suited as a universal definition of the Kondo energy.
When a single resonator is coupled to a continuous spectrum one obtains a resonance of finite half-width. Such a resonance is known in many fields of physics. The Friedel resonance is an example where a d-impurity is dissolved in a simple metal. If two resonators are coupled to the continuous spectrum the resonances interfere. For identical coupling and frequencies one obtains two effective resonances. The effective coupling of one of them to the continuum can be tuned to zero yielding a delta-like resonance.
Although the Kondo effect and the Kondo ground state of a magnetic impurity have been investigated for more than forty years it was until recently difficult if not impossible to calculate spatial properties of the ground state. In particular the calculation of the spatial distribution of the so-called Kondo cloud or even its existence have been elusive. In recent years a new method has been introduced to investigate the properties of magnetic impurities, the FAIR method, where the abbreviation stands for Friedel Artificially Inserted Resonance. The FAIR solution of the Friedel-Anderson and the Kondo impurity problems consists of only eight or four Slater states. Because of its compactness the spatial electron density and polarization can be easily calculated. In this article a short review of the method is given. A comparison with results from the large N-approximation, the Numerical Renormalization Group theory and other methods shows excellent agreement. The FAIR solution yields (for the first time) the electronic polarization in the Kondo cloud.
By applying a magnetic field whose Zeeman energy exceeds the Kondo energy by an order of magnitude the ground state of the Friedel-Anderson impurity is a magnetic state. In recent years the author introduced the Friedel Artificially Inserted Resonance (FAIR) method to investigate impurity properties. Within this FAIR approach the magnetic ground state is derived. Its full excitation spectrum and the composition of the excitations is calculated and numerically evaluated. From the excitation spectrum the electron density of states is calculated. Majority and minority d-resonances are obtained. The width of the resonances is about twice as wide as the mean field theory predicts. This broadening is due to the fact that any change of the occupation of the d-state in one spin band changes the eigenstates in the opposite spin band and causes transitions in both spin bands. This broadening reduces the height of the resonance curve and therefore the density of states by a factor of two. This yields an intuitive understanding for a previous result of the FAIR approach that the critical value of the Coulomb interaction for the formation of a magnetic moment is twice as large as the mean field theory predicts.
The electronic properties of a Kondo impurity are investigated in a magnetic field using linear response theory. The distribution of electrical charge and magnetic polarization are calculated in real space. The (small) magnetic field does not change the charge distribution. However, it unmasks the Kondo cloud. The (equal) weight of the d-electron components with their magnetic moment up and down is shifted and the compensating s-electron clouds don't cancel any longer (a requirement for an experimental detection of the Kondo cloud). In addition to the net magnetic polarization of the conduction electrons an oscillating magnetic polarization with a period of half the Fermi wave length is observed. However, this oscillating magnetic polarization does not show the long range behavior of Rudermann-Kittel-Kasuya-Yosida oscillations because the oscillations don't extend beyond the Kondo radius. They represent an internal electronic structure of the Kondo impurity in a magnetic field. PACS: 75.20.Hr, 71.23.An, 71.27.+a
In the presence of a magnetic impurity the spin-up and down band states are modified differently by the impurity. If the multi-electron scalar product (MESP) between the occupied spin-up and down states approaches zero then this defines an orthogonality catastrophe. In the present paper the MESP is investigated for the FAIR (Friedel Artificial Iserted Resonance) solution for a Friedel-Anderson impurity. A basis of Wilson states is used. The MESP is numerically determined for the (enforced) magnetic, the singlet, and the triplet states as a function of the number N of Wilson states. The magnetic and the triplet state show an exponentially decreasing MESP as a function of N. Surprisingly it is not the number of states which causes this decrease. It is instead the energy separation of the highest occupied state from the Fermi energy which determines the reduction of the MESP. In the singlet state the ground-state requires a finite MESP to optimize its energy. As a consequence there is no orthogonality catastrophe. The MESP approaches a saturation value as function of N.
Two atomic layers of Ni condensed onto Pb films behave, according to anomalous Hall effect measurements, as magnetic dead layers. However, the Ni lowers the superconducting T_{c} of the Pb film. This has lead to the conclusion that the Ni layers are still very weakly magnetic. In the present paper the electron dephasing due to the Ni has been measured by weak localization. The dephasing is smaller by a factor 100 than the pair-breaking. This proves that the T_{c}-reduction in the PbNi films is not due magnetic Ni moments.
Recently Affleck et al. derived the existence of Friedel oscillations in the presence of a Kondo impurity. They supported their analytic derivation by numerical calculations using Wilson's renormalization approach (NRG). In this paper the size of the Friedel oscillations is calculated with the FAIR method (Friedel Artificially Inserted Resonance) which has been recently developed. The results of NRG and FAIR are compared. The development of Friedel oscillations with a phase shift of pi/2 outside of the Kondo radius is confirmed.
The internal s-electron polarization in the ground state of a Friedel-Anderson and a Kondo impurity is calculated. The Wilson basis with exponentially fine energies close to the Fermi level is used and expressed in terms of free electron waves with linear energy-momentum dispersion. It is found that in the singlet state the magnetic moment of the d-electron is screened by an s-electron cloud. The linear extension of the cloud is inversely proportional to the Kondo energy. When the singlet state is destroyed the polarization cloud is absent.
In his renormalization paper of the Kondo effect Wilson replaced the full band of s-electrons by a small number of ''Wilson states''. He started from a rather artificial symmetric band with constant density of states and constant interaction with the impurity. It is shown in the present paper that with a minor modification the Wilson states are optimally suited to treat the interaction of an impurity with an arbitrary s-band. Each Wilson state represents electrons of a whole energy range. It carries the interaction of all these electrons with the impurity. All the other electron states in this energy range have zero interaction with the impurity and are neglected in the calculation. The resulting error is minor. As an example the singlet-triplet excitation energy of a Kondo impurity is numerically calculated for a tight-binding band with a strongly energy dependent density of states.
A compact approximate groundstate of the Kondo problem is introduced. It consists of four Slater states. The spin up and down states of the localized d-impurity are paired with two localized s-electron states of opposite spin. All the remaining s-electron states are rearranged forming two new optimal orthonormal bases. Through a rotation in Hilbert space the two localized states (and the rest of the bases) are optimized by minimizing the energy expectation value. The ground-state energy E and the singlet-triplet excitation energy dE are calculated numerically. Although the two energies can differ by a factor of 1000, they are obtained simultaneously. The singlet-triplet excitation energy dE is proportional to exp[-1/2Jg] and quite close to the Kondo temperature k_BT_K. The cases for anti-ferromagnetic (J>0) and ferromagnetic (J<0) coupling are investigated.
Cobalt nanowires with a diameter in the range between 50 to 100nm can be prepared as single-crystal wires with the easy axis (the c-axis) perpendicular to the wire axis. The competition between the crystal anisotropy and demagnetization energy frustrates the magnetization direction. A periodic modulation of the angle between M and the wire axis yields a lower energy.
A strong electron-phonon interaction in a metal increases the electron density of states in the vicinity of the Fermi energy dramatically. This phenomenon is called electron-phonon mass enhancement. In this paper the question is investigated whether the mass enhancement can be manipulated in multi-layers of two metals with strong and weak electron-phonon interaction. A rich behavior is observed for different thickness ranges of the layers. For thin layers one observes a rather homogeneous averaged enhancement. However, for an intermediate thickness range the mass enhancement is highly anisotropic, i.e. direction dependent, as well as position dependent. For large layer thicknesses one obtains the bulk behavior for each metal.